MATH 108 - ELEMENTARY AND
INTERMEDIATE ALGEBRA -
Differentiation of Trigonometric
Functions
Question Bank - Set 4
Liberty University
Question 1
Question
Find the derivative of the function f(x) = sin(2x) cos(3x).
Solution
Step 1: We will use the product rule to find the derivative of f(x). The product
rule states that if f(x) = g(x)h(x), then f′(x) = g′(x)h(x) + g(x)h′(x).
Step 2: Let g(x) = sin(2x) and h(x) = cos(3x). Then, we have g′(x) =
2 cos(2x) and h′(x) = −3 sin(3x).
Step 3: Now, apply the product rule to find f′(x):
f′(x) = g′(x)h(x) + g(x)h′(x)
= (2 cos(2x))(cos(3x)) + (sin(2x))(−3 sin(3x))
= 2 cos(2x) cos(3x)−3 sin(2x) sin(3x).
Therefore, the derivative of the function f(x) = sin(2x) cos(3x) is f′(x) =
2 cos(2x) cos(3x)−3 sin(2x) sin(3x).
Question 2
Question
Find the derivative of the function f(x) = tan2(x)−sin(x) cos(x).
Solution
To find the derivative of f(x) = tan2(x)−sin(x) cos(x), we will differentiate
term by term using the rules of differentiation.
Step 1: Let’s find the derivative of tan2(x).
d
dx (tan2(x)) = 2 tan(x) sec2(x)
Step 2: Now, let’s find the derivative of sin(x) cos(x).
d
dx (sin(x) cos(x)) = cos2(x)−sin2(x)
Step 3: Putting it all together, the derivative of f(x) = tan2(x)−sin(x) cos(x)
is:
d
dx (f(x)) = 2 tan(x) sec2(x)−(cos2(x)−sin2(x))
= 2 tan(x) sec2(x)−cos2(x) + sin2(x)
Therefore, the derivative of f(x) = tan2(x)−sin(x) cos(x) is 2 tan(x) sec2(x)−
cos2(x) + sin2(x).
Question 3
Question
Find the derivative of y= sin(2x) cos(3x) with respect to x.
Solution
Step 1: Apply the product rule for differentiation, which states that the deriva-
tive of a product of two functions is the derivative of the first function times
the second function, plus the first function times the derivative of the sec-
ond function. Step 2: Let u= sin(2x) and v= cos(3x). Step 3: Find
the derivatives of uand vwith respect to x. Step 4: du
dx = 2 cos(2x) and
dv
dx =−3 sin(3x). Step 5: Apply the product rule: dy
dx =udv
dx +vdu
dx . Step
6: Substitute the values of u,v,du
dx , and dv
dx into the formula above. Step
7: dy
dx = sin(2x)·(−3 sin(3x)) + cos(3x)·2 cos(2x). Step 8: Simplify the ex-
pression in Step 7 by expanding and using trigonometric identities. Step 9:
dy
dx =−3 sin(2x) sin(3x) + 2 cos(3x) cos(2x). Step 10: Use the trigonometric
identity sin(A) sin(B) = 1
2[cos(A−B)−cos(A+B)] to simplify the expression
further. Step 11: dy
dx =−3
2[cos(2x+ 3x)−cos(2x−3x)] + 2 cos(3x) cos(2x).
Step 12: Simplify the angles in the cosine terms. Step 13: dy
dx =−3
2[cos(5x)−
cos(−x)] + 2 cos(3x) cos(2x). Step 14: Note that cos(−x) = cos(x). Step 15:
dy
dx =−3
2[cos(5x)−cos(x)] + 2 cos(3x) cos(2x). Therefore, the derivative of y=
sin(2x) cos(3x) with respect to xis dy
dx =−3
2[cos(5x)−cos(x)]+2 cos(3x) cos(2x).
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Question 4
Question
Find the derivative of the function f(x) = sin(x) cos(x).
Solution
Step 1: Apply the product rule, which states that the derivative of a product
of two functions is the derivative of the first function times the second function,
plus the first function times the derivative of the second function. Let u= sin(x)
and v= cos(x). Then, we have
f(x) = u·v
Step 2: Calculate the derivatives of uand v.
Derivative of u= sin(x):
du
dx = cos(x)
Derivative of v= cos(x):
dv
dx =−sin(x)
Step 3: Apply the product rule to find the derivative of f(x).
f′(x) = u′·v+u·v′
Step 4: Substitute the derivatives of uand vinto the formula for f′(x).
f′(x) = cos(x)·cos(x) + sin(x)·(−sin(x))
Step 5: Simplify the expression.
f′(x) = cos2(x)−sin2(x)
Step 6: Recall the trigonometric identity cos2(x)−sin2(x) = cos(2x). There-
fore, the derivative of f(x) = sin(x) cos(x) is:
f′(x) = cos(2x)
Question 5
Question
Find the derivative of the function f(x) = sin2(x) cos(x).
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Solution
Step 1: Use the product rule to differentiate the function f(x). Step 2: Apply
the chain rule to differentiate sin2(x). Step 3: Simplify the expression to find
the derivative of f(x).
Step 1: The product rule states that if u(x) and v(x) are differentiable
functions, then the derivative of their product is given by:
(uv)′=u′v+uv′
In this case, u(x) = sin2(x) and v(x) = cos(x). Therefore, the derivative of f(x)
is:
f′(x) = (sin2(x))′cos(x) + sin2(x)(cos(x))′
Step 2: To differentiate sin2(x), we need to apply the chain rule. Let
u= sin(x), then d
dx (sin2(x)) = d
dx (sin(u)2) = 2 sin(x) cos(x).
Step 3: Substitute the derivatives back into the expression for f′(x):
f′(x) = 2 sin(x) cos(x) cos(x) + sin2(x)(−sin(x))
Simplify the expression:
f′(x) = 2 sin(x) cos2(x)−sin3(x)
So, the derivative of the function f(x) = sin2(x) cos(x) is f′(x) = 2 sin(x) cos2(x)−
sin3(x).
Question 6
Question
Let f(x) = sin(x) cos(x). Find f′(x).
Solution
Step 1: We will use the product rule to differentiate f(x) = sin(x) cos(x).
Step 2: Recall that the product rule states that if f(x) = u(x)v(x), then
f′(x) = u′(x)v(x) + u(x)v′(x).
Step 3: Let u(x) = sin(x) and v(x) = cos(x).
Step 4: Find u′(x) and v′(x).
u′(x) = d
dx (sin(x))
= cos(x)
v′(x) = d
dx (cos(x))
=−sin(x)
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Step 5: Apply the product rule to find f′(x).
f′(x) = u′(x)v(x) + u(x)v′(x)
= cos(x)·cos(x) + sin(x)·(−sin(x))
= cos2(x)−sin2(x)
= cos(2x)
Step 6: Therefore, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos(2x).
Question 7
Question
Find the derivative of f(x) = sin(2x) cos(3x).
Solution
Step 1: Apply the product rule to differentiate f(x).
f′(x) = sin(2x)·d
dx (cos(3x))+d
dx (sin(2x)) ·cos(3x)
Step 2: Compute the derivative of cos(3x) and sin(2x).
d
dx (cos(3x)) = −3 sin(3x)
d
dx (sin(2x)) = 2 cos(2x)
Step 3: Substitute the derivatives back into f′(x).
f′(x) = (sin(2x)(−3 sin(3x))) + (2 cos(2x)·cos(3x))
=−3 sin(2x) sin(3x) + 2 cos(2x) cos(3x)
Therefore, the derivative of f(x) = sin(2x) cos(3x) is f′(x) = −3 sin(2x) sin(3x)+
2 cos(2x) cos(3x).
Question 8
Question
Find the derivative of the function f(x) = csc(2x) + cot(3x).
5
Solution
Step 1: Recall the derivatives of the trigonometric functions. The derivatives of
the six trigonometric functions are as follows:
d
dx (sin(x)) = cos(x),d
dx (cos(x)) = −sin(x),
d
dx (tan(x)) = sec2(x),d
dx (csc(x)) = −csc(x) cot(x),
d
dx (sec(x)) = sec(x) tan(x),d
dx (cot(x)) = −csc2(x).
Step 2: Find the derivative of f(x) = csc(2x) + cot(3x) using the above
derivatives.
d
dx (csc(2x)) = −csc(2x) cot(2x)·2 = −2 csc(2x) cot(2x)
d
dx (cot(3x)) = −csc2(3x)·3 = −3 csc2(3x)
Step 3: Combine the derivatives to find the derivative of f(x).
f′(x) = −2 csc(2x) cot(2x)−3 csc2(3x)
Therefore, the derivative of f(x) = csc(2x)+cot(3x) is f′(x) = −2 csc(2x) cot(2x)−3 csc2(3x) .
Question 9
Question
Find the derivative of the function f(x) = cos2(x) sin(2x).
Solution
Step 1: Use the product rule to differentiate f(x) = cos2(x) sin(2x). Step 2:
Let u= cos2(x) and v= sin(2x). Step 3: Compute u′=−2 cos(x) sin(x) and
v′= 2 cos(2x). Step 4: Apply the product rule f′(x) = u′v+uv′. Step 5:
Substitute the values of u,u′,vand v′into the product rule formula. Step 6:
Simplify the expression to find the final derivative of f(x).
Question 10
Question
Find the derivative of the following function:
f(x) = sin(x) + cos(x)
sin(x)−cos(x)
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Solution
To find the derivative of the given function, we need to apply the quotient rule.
Let u(x) = sin(x) + cos(x) and v(x) = sin(x)−cos(x).
Step 1: Determine u′(x) and v′(x).
For u(x) = sin(x) + cos(x):
u′(x) = cos(x)−sin(x)
For v(x) = sin(x)−cos(x):
v′(x) = cos(x) + sin(x)
Step 2: Apply the quotient rule to find f′(x).
f′(x) = u′(x)v(x)−v′(x)u(x)
(v(x))2
Step 3: Substitute u′(x), v′(x), u(x), and v(x) into the quotient rule for-
mula.
f′(x) = (cos(x)−sin(x))(sin(x)−cos(x)) −(cos(x) + sin(x))(sin(x) + cos(x))
(sin(x)−cos(x))2
=cos(x) sin(x)−cos2(x)−sin2(x) + sin(x) cos(x)−cos(x) sin(x)−sin2(x)−cos(x) sin(x)−sin(x) cos(x)
(sin(x)−cos(x))2
=−2 cos2(x)−2 sin2(x)
(sin(x)−cos(x))2
=−2(cos2(x) + sin2(x))
(sin(x)−cos(x))2
=−2
(sin(x)−cos(x))2
Therefore, the derivative of the function f(x) = sin(x)+cos(x)
sin(x)−cos(x)is f′(x) =
−2
(sin(x)−cos(x))2.
Question 11
Question
Find the derivative of y= sin(2x) + cos(3x).
Solution
Step 1: Use the chain rule to find the derivative of sin(2x).
d
dx [sin(2x)] = cos(2x)·d
dx (2x)
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Step 2: Simplify the derivative of sin(2x).
d
dx [sin(2x)] = 2 cos(2x)
Step 3: Use the chain rule to find the derivative of cos(3x).
d
dx [cos(3x)] = −sin(3x)·d
dx (3x)
Step 4: Simplify the derivative of cos(3x).
d
dx [cos(3x)] = −3 sin(3x)
Step 5: Combine the derivatives of sin(2x) and cos(3x) to find the derivative
of y.
y′= 2 cos(2x)−3 sin(3x)
Therefore, the derivative of y= sin(2x)+cos(3x) is y′= 2 cos(2x)−3 sin(3x).
Question 12
Question
Differentiate the function f(x) = tan(x)
cos(x).
Solution
Step 1: Rewrite the function using trigonometric identities. Step 2: Differentiate
using the quotient rule.
Step 1: We rewrite the function f(x) = tan(x)
cos(x)using trigonometric identities:
f(x) = sin(x)
cos(x)·1
cos(x)
f(x) = sin(x)
cos2(x)
Step 2: Now we differentiate f(x) using the quotient rule. If f(x) = g(x)
h(x),
then f′(x) = g′(x)h(x)−g(x)h′(x)
(h(x))2. Applying the quotient rule to f(x) = sin(x)
cos2(x),
we have:
f′(x) = cos2(x) cos(x)−sin(x)(−2 cos(x) sin(x))
cos4(x)
f′(x) = cos3(x) + 2 sin2(x) cos(x)
cos4(x)
f′(x) = cos(x)(1 + 2 sin2(x))
cos4(x)
8
f′(x) = cos(x)(1 + 2 sin2(x))
cos4(x)
f′(x) = cos(x)(1 + 2(1 −cos2(x)))
cos4(x)
f′(x) = cos(x)(1 + 2 −2 cos2(x))
cos4(x)
f′(x) = cos(x)(3 −2 cos2(x))
cos4(x)
f′(x) = 3 cos(x)−2 cos3(x)
cos4(x)
f′(x) = 3−2 cos2(x)
cos3(x)
Therefore, the derivative of f(x) = tan(x)
cos(x)is 3−2 cos2(x)
cos3(x).
Question 13
Question
Find the derivative of f(x) = sin(2x) cos(3x).
Solution
Step 1: Apply the product rule, which states that if f(x) = u(x)v(x), then
f′(x) = u′v+uv′.
Step 2: Let u(x) = sin(2x) and v(x) = cos(3x). Then, we need to find u′(x)
and v′(x).
Step 3: Find u′(x) by applying the chain rule. Since the derivative of sin(u)
is cos(u)u′, we have u′(x) = cos(2x)·2.
Step 4: Find v′(x) by applying the chain rule. Since the derivative of cos(u)
is −sin(u)u′, we have v′(x) = −sin(3x)·3.
Step 5: Now, apply the product rule to find f′(x):
f′(x) = u′v+uv′= (cos(2x)·2)(cos(3x)) + (sin(2x))(−sin(3x)·3)
Step 6: Simplify the expression to get the final answer for f′(x).
f′(x) = 2 cos(2x) cos(3x)−3 sin(2x) sin(3x)
Question 14
Question
Find the derivative of the function f(x) = sin(2x) cos(3x).
9
Solution
To find the derivative of f(x) = sin(2x) cos(3x), we will use the product rule.
Step 1: Apply the product rule. Let u= sin(2x) and v= cos(3x). Then,
using the product rule f′(x) = u′v+uv′, we have:
f′(x) = (2 cos(2x))(cos(3x)) + (sin(2x))(−3 sin(3x))
Step 2: Simplify the derivative. Substituting u= sin(2x), u′= 2 cos(2x),
v= cos(3x), and v′=−3 sin(3x) into the formula:
f′(x) = 2 cos(2x) cos(3x)−3 sin(2x) sin(3x)
Thus, the derivative of the function f(x) = sin(2x) cos(3x) is f′(x) =
2 cos(2x) cos(3x)−3 sin(2x) sin(3x).
Question 15
Question
Find the derivative of y= cos2(3x) sin(2x) with respect to x.
Solution
Step 1: Apply the product rule to differentiate the function. Let u= cos2(3x)
and v= sin(2x). Then, y=uv, and the derivative of ycan be found using the
formula dy
dx =udv
dx +vdu
dx .
Step 2: Find du
dx and dv
dx .
du
dx =d
dx (cos2(3x)) = 2 cos(3x)(−3 sin(3x)) = −6 cos(3x) sin(3x)
dv
dx =d
dx (sin(2x)) = 2 cos(2x)
Step 3: Substitute u,v,du
dx , and dv
dx into the product rule formula.
dy
dx = (cos2(3x))(2 cos(2x)) + (sin(2x))(−6 cos(3x) sin(3x))
Step 4: Simplify the expression.
dy
dx = 2 cos3(3x) cos(2x)−6 sin(2x) cos(3x) sin(3x)
Therefore, the derivative of y= cos2(3x) sin(2x) with respect to xis 2 cos3(3x) cos(2x)−
6 sin(2x) cos(3x) sin(3x).
10
Question 16
Question
Find the derivative of the function f(x) = sin(2x) + cos(3x).
Solution
Step 1: We will differentiate each term of the function separately using the rules
for differentiating trigonometric functions. Step 2: Recall that the derivative
of sin(u) is cos(u) multiplied by the derivative of u. Similarly, the derivative of
cos(v) is −sin(v) multiplied by the derivative of v.
Therefore, the derivative of f(x) can be found as follows:
f′(x) = d
dx [sin(2x)] + d
dx [cos(3x)]
Step 3: Applying the chain rule, we get:
f′(x) = cos(2x)·d
dx (2x)−sin(3x)·d
dx (3x)
Step 4: Simplifying, we have:
f′(x) = 2 cos(2x)−3 sin(3x)
Step 5: Therefore, the derivative of the function f(x) = sin(2x) + cos(3x) is
f′(x) = 2 cos(2x)−3 sin(3x).
Question 17
Question
Differentiate the function f(x) = sinx2+ cos(2x) with respect to x.
Solution
Step 1: Apply the chain rule to differentiate sinx2.
d
dx (sinx2) = cosx2·d
dx (x2).
Step 2: Simplify the derivative.
d
dx (sinx2) = cosx2·2x= 2xcosx2.
Step 3: Apply the chain rule to differentiate cos(2x).
d
dx (cos(2x)) = −sin(2x)·d
dx (2x).
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Step 4: Simplify the derivative.
d
dx (cos(2x)) = −sin(2x)·2 = −2 sin(2x).
Step 5: Combine the derivatives to find f′(x).
f′(x) = 2xcosx2−2 sin(2x).
Therefore, the derivative of f(x) = sinx2+ cos(2x) with respect to xis
f′(x) = 2xcosx2−2 sin(2x) .
Question 18
Question
Find the derivative of the function f(x) = cos(2x)·sin(3x).
Solution
Step 1: Apply the product rule to differentiate the product of cos(2x) and
sin(3x). Step 2: Let u= cos(2x) and v= sin(3x). Step 3: Find u′and v′by
differentiating uand vwith respect to x. Step 4: Calculate f′(x) by applying
the product rule formula f′(x) = u′v+uv′. Step 5: Simplify the derivative
expression to get the final answer.
Step 1: Apply the product rule:
f′(x) = (cos(2x))′·sin(3x) + cos(2x)·(sin(3x)′)
Step 2: Let u= cos(2x) and v= sin(3x).
Step 3: Find u′and v′:
u′=−2 sin(2x), v′= 3 cos(3x)
Step 4: Substitute u′and v′into the product rule formula:
f′(x)=(−2 sin(2x)·sin(3x)) + (cos(2x)·3 cos(3x))
Step 5: Simplify the expression:
f′(x) = −2 sin(2x) sin(3x) + 3 cos(2x) cos(3x)
Therefore, the derivative of the function f(x) = cos(2x)·sin(3x) is −2 sin(2x) sin(3x)+
3 cos(2x) cos(3x).
Question 19
Question
Find the derivative of the function f(x) = sin2(x) cos(x).
12
Solution
Step 1: We will use the product rule to find the derivative of the given function
f(x). Step 2: Recall that the product rule states: (uv)′=u′v+uv′, where uand
vare differentiable functions of x. Step 3: Let u(x) = sin2(x) and v(x) = cos(x).
Step 4: Find the derivatives of u(x) and v(x):
u′(x) = 2 sin(x) cos(x)
v′(x) = −sin(x)
Step 5: Apply the product rule to find f′(x):
f′(x) = u′v+uv′
= (2 sin(x) cos(x))(cos(x)) + (sin2(x))(−sin(x))
= 2 sin(x) cos2(x)−sin3(x)
Step 6: Therefore, the derivative of f(x) = sin2(x) cos(x) is f′(x) = 2 sin(x) cos2(x)−sin3(x) .
Question 20
Question
Let f(x) = cos2(2x). Find f′(x).
Solution
Step 1: Apply the chain rule.
f′(x) = d
dx (cos2(2x))
= 2 cos(2x)·−sin(2x)·d
dx (2x)
Step 2: Simplify by finding d
dx (2x).
= 2 cos(2x)·(−sin(2x)·2)
Step 3: Simplify the expression.
=−4 cos(2x) sin(2x)
Step 4: Use the double-angle formula sin(2x) = 2 sin(x) cos(x).
=−4 cos(2x)·2 sin(x) cos(x)
Step 5: Apply the double-angle formula cos(2x) = 2 cos2(x)−1.
=−4(2 cos2(x)−1) ·2 sin(x) cos(x)
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Step 6: Simplify and expand the expression.
=−8 cos2(x) sin(x) cos(x) + 8 sin(x) cos(x)
Step 7: Use the double-angle formula sin(2x) = 2 sin(x) cos(x) to simplify.
=−8 cos2(x) sin(x) cos(x) + 8 sin(x) cos(x)
Step 8: Finally, simplify the expression.
=−8 cos2(x) sin(x) + 8 sin(x) cos(x)
Question 21
Question
Find the derivative of f(x) = sin2(x) cos(2x).
Solution
To find the derivative of f(x), we will use the product rule and chain rule of
differentiation.
Step 1: Apply the product rule. Let u= sin2(x) and v= cos(2x). Then,
f′(x) = u′v+uv′
Step 2: Find u′and v′.
Derivative of u= sin2(x):
u′= 2 sin(x) cos(x)
Derivative of v= cos(2x):
v′=−2 sin(2x)
Step 3: Substitute u′and v′back into the product rule formula.
f′(x) = (2 sin(x) cos(x))(cos(2x)) + (sin2(x))(−2 sin(2x))
Step 4: Simplify the expression.
f′(x) = 2 sin(x) cos(x) cos(2x)−2 sin2(x) sin(2x)
Step 5: Use trigonometric identities to further simplify the expression. For
example, use the double angle identities:
sin(2x) = 2 sin(x) cos(x)
cos(2x) = cos2(x)−sin2(x)
Step 6: Substitute the identities into the derivative expression and simplify
further if needed.
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Question 22
Question
Find the derivative of the function f(x) = cos2(3x) + sin2(2x).
Solution
Step 1: Recall the trigonometric identity cos2(x) + sin2(x) = 1 for any angle x.
Step 2: Rewrite the function f(x) using this identity: f(x) = 1.
Step 3: The derivative of a constant function is 0.
Step 4: Therefore, the derivative of f(x) = cos2(3x) + sin2(2x) is 0 .
Question 23
Question
Find the derivative of the function f(x) = sin2(3x) + cos2(5x).
Solution
Step 1: Use the identity sin2(x) + cos2(x) = 1 to simplify the function. Step
2: Differentiate the simplified function using the chain rule for trigonometric
functions. Step 3: Rewrite the derivative in terms of cosine functions if needed.
Step 1: First, let’s simplify the given function f(x) using the trigonometric
identity sin2(x) + cos2(x) = 1:
f(x) = sin2(3x) + cos2(5x)
= 1 + 1
= 2
So, f(x) = 2.
Step 2: Now, let’s find the derivative of the function f(x) = 2 with respect
to x:d
dx f(x) = d
dx 2
= 0
Therefore, the derivative of f(x) = sin2(3x) + cos2(5x) is 0.
Step 3: The derivative of a constant function is 0, so the derivative of f(x)
is indeed 0.
Question 24
Question
Find the derivative of the function f(x) = sin(3x) cos(2x).
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Solution
To find the derivative of f(x) = sin(3x) cos(2x), we will use the product rule,
which states that if h(x) = g(x)·k(x), then h′(x) = g′(x)·k(x) + g(x)·k′(x).
Step 1: Identify g(x) and k(x) in f(x) = sin(3x) cos(2x).
Here, g(x) = sin(3x) and k(x) = cos(2x).
Step 2: Find g′(x) and k′(x).
Using the chain rule, g′(x) = 3 cos(3x) and k′(x) = −2 sin(2x).
Step 3: Apply the product rule.
The derivative of f(x) is:
f′(x) = g′(x)·k(x) + g(x)·k′(x)
= 3 cos(3x)·cos(2x) + sin(3x)·(−2 sin(2x))
= 3 cos(3x) cos(2x)−2 sin(3x) sin(2x).
Therefore, the derivative of f(x) = sin(3x) cos(2x) is f′(x) = 3 cos(3x) cos(2x)−
2 sin(3x) sin(2x).
Question 25
Question
Find the derivative of the function f(x) = sin(x) cos(x).
Solution
Step 1: Apply the product rule.
Let u= sin(x) and v= cos(x). Then, the product rule states that the
derivative of u·vis given by:
(u·v)′=u′v+uv′
Step 2: Find u′and v′.
Since d
dx (sin(x)) = cos(x) and d
dx (cos(x)) = −sin(x), we get:
u′= cos(x)
v′=−sin(x)
Step 3: Substitute u′and v′into the product rule formula.
So, the derivative of f(x) = sin(x) cos(x) can be found using the product
rule as:
f′(x) = (sin(x)·cos(x))′= (cos(x)·cos(x) + sin(x)· − sin(x))
f′(x) = cos2(x)−sin2(x)
Therefore, the derivative of the function f(x) = sin(x) cos(x) is f′(x) =
cos2(x)−sin2(x).
16
Question 26
Question
Find the derivative of the function f(x) = sin(2x) cos(3x).
Solution
Step 1: Apply the product rule to find the derivative of the function f(x) =
sin(2x) cos(3x). Step 2: Let u= sin(2x) and v= cos(3x). Step 3: Find u′
and v′. Step 4: Apply the product rule: f′(x) = u′v+uv′. Step 5: Substitute
back u,v,u′, and v′into the formula. Step 6: Simplify the expression for the
derivative.
Question 27
Question
Find the derivative of the function f(x) = 3 sin(2x) + cos(x).
Solution
To find the derivative of f(x) = 3 sin(2x)+cos(x), we will use the differentiation
rules for trigonometric functions.
Step 1: Find f′(x) by applying the derivative rules.
Recall the derivative rules: - d
dx [sin(ax)] = acos(ax) - d
dx [cos(ax)] = −asin(ax)
Using these rules, we find:
f′(x) = d
dx [3 sin(2x)] + d
dx [cos(x)]
= 3(2 cos(2x)) −sin(x)
= 6 cos(2x)−sin(x).
Therefore, the derivative of f(x) = 3 sin(2x)+cos(x) is f′(x) = 6 cos(2x)−sin(x) .
Question 28
Question
Find the derivative of the function f(x) = sin2(x) cos2(x).
17
Solution
Step 1: Use the product rule to differentiate f(x) = sin2(x) cos2(x).
f′(x) = d
dx (sin2(x)) ·cos2(x) + sin2(x)·d
dx (cos2(x))
= 2 sin(x)·cos(x)·cos2(x) + sin2(x)·2 cos(x)·(−sin(x))
= 2 sin(x)·cos(x)·cos2(x)−2 sin(x)·cos(x)·sin2(x)
Step 2: Factor out a common factor of 2 sin(x)·cos(x).
f′(x) = 2 sin(x)·cos(x)·(cos2(x)−sin2(x))
= 2 sin(x)·cos(x)·(cos(2x))
Therefore, the derivative of the function f(x) = sin2(x) cos2(x) is f′(x) =
2 sin(x)·cos(x)·cos(2x).
Question 29
Question
Find the derivative of the function f(x) = sec(x) + tan(x).
Solution
Step 1: We will differentiate each term separately using the rules of differen-
tiation. Step 2: Recall that d
dx (sec(x)) = sec(x) tan(x). Step 3: Similarly,
d
dx (tan(x)) = sec2(x). Step 4: Therefore, the derivative of f(x) = sec(x)+tan(x)
is d
dx (f(x)) = d
dx (sec(x)) + d
dx (tan(x))
Step 5: Substituting the derivatives we found earlier, we get
d
dx (f(x)) = sec(x) tan(x) + sec2(x)
Step 6: Thus, the derivative of the function f(x) = sec(x)+tan(x) is sec(x) tan(x)+
sec2(x).
Question 30
Question
Find the derivative of the function f(x) = sin(x) cos(x).
18
Solution
To find the derivative of f(x) = sin(x) cos(x), we will use the product rule,
which states that the derivative of a product of two functions is the first function
times the derivative of the second function, plus the second function times the
derivative of the first function.
Step 1: Identify the functions uand v. Let u(x) = sin(x) and v(x) = cos(x).
Step 2: Find u′(x) and v′(x). Using the derivatives of sine and cosine
functions, we have: u′(x) = cos(x) and v′(x) = −sin(x).
Step 3: Apply the product rule. The derivative of f(x) = sin(x) cos(x) is:
f′(x) = u′v+uv′
Substitute u′,v′,u, and vinto the formula:
f′(x) = cos(x) cos(x) + sin(x)(−sin(x))
Step 4: Simplify the expression. Simplify the expression to get the final
answer:
f′(x) = cos2(x)−sin2(x)
Therefore, the derivative of the function f(x) = sin(x) cos(x) is f′(x) =
cos2(x)−sin2(x).
Question 31
Question
Find the derivative of f(x) = sin2(2x) + cos(3x).
Solution
Step 1: Apply the chain rule to differentiate sin2(2x).
Let u= sin(2x) =⇒du
dx = 2 cos(2x)
⇒d
dx (sin2(2x)) = 2 sin(2x) cos(2x)
Step 2: Differentiate cos(3x) using the chain rule.
d
dx (cos(3x)) = −3 sin(3x)
Step 3: Combining the derivatives, we get the derivative of f(x).
d
dx (f(x)) = 2 sin(2x) cos(2x)−3 sin(3x)
Therefore, the derivative of f(x) = sin2(2x) + cos(3x) is 2 sin(2x) cos(2x)−
3 sin(3x).
19
Question 32
Question
Find the derivative of f(x) = sin(2x) cos(3x) with respect to x.
Solution
To find the derivative of f(x) = sin(2x) cos(3x) with respect to x, we will use the
product rule of differentiation: (uv)′=u′v+uv′, where uand vare functions
of x.
Step 1: Let u= sin(2x) and v= cos(3x). Then we have:
u′= 2 cos(2x) and v′=−3 sin(3x)
Step 2: Apply the product rule:
(f(x))′= (uv)′
=u′v+uv′
= (2 cos(2x))(cos(3x)) + (sin(2x))(−3 sin(3x))
= 2 cos(2x) cos(3x)−3 sin(2x) sin(3x)
Hence, the derivative of f(x) = sin(2x) cos(3x) with respect to xis 2 cos(2x) cos(3x)−
3 sin(2x) sin(3x).
Question 33
Question
Find the derivative of the function f(x) = cos(x)−2 sin(x)
tan(x).
Solution
To find the derivative of the given function, f(x) = cos(x)−2 sin(x)
tan(x), we will use
the quotient rule.
Step 1: Apply the quotient rule. Let u(x) = cos(x)−2 sin(x) and v(x) =
tan(x). The quotient rule states that the derivative of u(x)
v(x)is given by:
d
dx u(x)
v(x)=v(x)u′(x)−u(x)v′(x)
(v(x))2
Step 2: Calculate the derivatives of u(x) and v(x). We have: u′(x) =
−sin(x)−2 cos(x) (derivative of cos(x)−2 sin(x))
v′(x) = sec2(x) (derivative of tan(x))
20
Step 3: Apply the quotient rule formula. Substitute u′(x), v(x), u(x), and
v′(x) into the formula:
f′(x) = tan(x)(−sin(x)−2 cos(x)) −(cos(x)−2 sin(x)) sec2(x)
(tan(x))2
Step 4: Simplify the expression.
f′(x) = −sin(x) tan(x)−2 cos(x) tan(x)−cos(x) sec2(x) + 2 sin(x) sec2(x)
tan2(x)
Step 5: Further simplify the expression. Using trigonometric identities, we
can simplify the expression further. For example,
tan(x) = sin(x)
cos(x)and sec2(x) = 1 + tan2(x)
Step 6: Final answer. After simplification, the derivative of f(x) is:
f′(x) = −3 sin(x)−2 cos(x)−cos(x)−tan(x)
cos2(x)
Question 34
Question
Find the derivative of the function f(x) = sin2(x) cos(x).
Solution
Step 1: Apply the product rule to the function f(x) = sin2(x) cos(x). Step 2:
Let u= sin2(x) and v= cos(x). Step 3: Find u′and v′. Step 4: Differentiate u
with respect to x:u′= 2 sin(x) cos(x). Step 5: Differentiate vwith respect to
x:v′=−sin(x). Step 6: Apply the product rule: f′(x) = u′v+uv′. Step 7:
Substitute u′,v′,u, and vinto the product rule formula. Step 8: Simplify the
expression to find the derivative of f(x).
Question 35
Question
Find the derivative of y= sin(3x) + cos(2x) with respect to x.
Solution
Step 1: Use the fact that d
dx (sin(ax)) = acos(ax) and d
dx (cos(ax)) = −asin(ax)
for any constant a.
21
Solution
To find the derivative of f(x) = tan2(x)−sin(x) cos(x), we will differentiate
term by term using the rules of differentiation.
Step 1: Let’s find the derivative of tan2(x).
d
dx (tan2(x)) = 2 tan(x) sec2(x)
Step 2: Now, let’s find the derivative of sin(x) cos(x).
d
dx (sin(x) cos(x)) = cos2(x)−sin2(x)
Step 3: Putting it all together, the derivative of f(x) = tan2(x)−sin(x) cos(x)
is:
d
dx (f(x)) = 2 tan(x) sec2(x)−(cos2(x)−sin2(x))
= 2 tan(x) sec2(x)−cos2(x) + sin2(x)
Therefore, the derivative of f(x) = tan2(x)−sin(x) cos(x) is 2 tan(x) sec2(x)−
cos2(x) + sin2(x).
Question 3
Question
Find the derivative of y= sin(2x) cos(3x) with respect to x.
Solution
Step 1: Apply the product rule for differentiation, which states that the deriva-
tive of a product of two functions is the derivative of the first function times
the second function, plus the first function times the derivative of the sec-
ond function. Step 2: Let u= sin(2x) and v= cos(3x). Step 3: Find
the derivatives of uand vwith respect to x. Step 4: du
dx = 2 cos(2x) and
dv
dx =−3 sin(3x). Step 5: Apply the product rule: dy
dx =udv
dx +vdu
dx . Step
6: Substitute the values of u,v,du
dx , and dv
dx into the formula above. Step
7: dy
dx = sin(2x)·(−3 sin(3x)) + cos(3x)·2 cos(2x). Step 8: Simplify the ex-
pression in Step 7 by expanding and using trigonometric identities. Step 9:
dy
dx =−3 sin(2x) sin(3x) + 2 cos(3x) cos(2x). Step 10: Use the trigonometric
identity sin(A) sin(B) = 1
2[cos(A−B)−cos(A+B)] to simplify the expression
further. Step 11: dy
dx =−3
2[cos(2x+ 3x)−cos(2x−3x)] + 2 cos(3x) cos(2x).
Step 12: Simplify the angles in the cosine terms. Step 13: dy
dx =−3
2[cos(5x)−
cos(−x)] + 2 cos(3x) cos(2x). Step 14: Note that cos(−x) = cos(x). Step 15:
dy
dx =−3
2[cos(5x)−cos(x)] + 2 cos(3x) cos(2x). Therefore, the derivative of y=
sin(2x) cos(3x) with respect to xis dy
dx =−3
2[cos(5x)−cos(x)]+2 cos(3x) cos(2x).
2
Question 4
Question
Find the derivative of the function f(x) = sin(x) cos(x).
Solution
Step 1: Apply the product rule, which states that the derivative of a product
of two functions is the derivative of the first function times the second function,
plus the first function times the derivative of the second function. Let u= sin(x)
and v= cos(x). Then, we have
f(x) = u·v
Step 2: Calculate the derivatives of uand v.
Derivative of u= sin(x):
du
dx = cos(x)
Derivative of v= cos(x):
dv
dx =−sin(x)
Step 3: Apply the product rule to find the derivative of f(x).
f′(x) = u′·v+u·v′
Step 4: Substitute the derivatives of uand vinto the formula for f′(x).
f′(x) = cos(x)·cos(x) + sin(x)·(−sin(x))
Step 5: Simplify the expression.
f′(x) = cos2(x)−sin2(x)
Step 6: Recall the trigonometric identity cos2(x)−sin2(x) = cos(2x). There-
fore, the derivative of f(x) = sin(x) cos(x) is:
f′(x) = cos(2x)
Question 5
Question
Find the derivative of the function f(x) = sin2(x) cos(x).
3
Solution
Step 1: Use the product rule to differentiate the function f(x). Step 2: Apply
the chain rule to differentiate sin2(x). Step 3: Simplify the expression to find
the derivative of f(x).
Step 1: The product rule states that if u(x) and v(x) are differentiable
functions, then the derivative of their product is given by:
(uv)′=u′v+uv′
In this case, u(x) = sin2(x) and v(x) = cos(x). Therefore, the derivative of f(x)
is:
f′(x) = (sin2(x))′cos(x) + sin2(x)(cos(x))′
Step 2: To differentiate sin2(x), we need to apply the chain rule. Let
u= sin(x), then d
dx (sin2(x)) = d
dx (sin(u)2) = 2 sin(x) cos(x).
Step 3: Substitute the derivatives back into the expression for f′(x):
f′(x) = 2 sin(x) cos(x) cos(x) + sin2(x)(−sin(x))
Simplify the expression:
f′(x) = 2 sin(x) cos2(x)−sin3(x)
So, the derivative of the function f(x) = sin2(x) cos(x) is f′(x) = 2 sin(x) cos2(x)−
sin3(x).
Question 6
Question
Let f(x) = sin(x) cos(x). Find f′(x).
Solution
Step 1: We will use the product rule to differentiate f(x) = sin(x) cos(x).
Step 2: Recall that the product rule states that if f(x) = u(x)v(x), then
f′(x) = u′(x)v(x) + u(x)v′(x).
Step 3: Let u(x) = sin(x) and v(x) = cos(x).
Step 4: Find u′(x) and v′(x).
u′(x) = d
dx (sin(x))
= cos(x)
v′(x) = d
dx (cos(x))
=−sin(x)
4
Step 5: Apply the product rule to find f′(x).
f′(x) = u′(x)v(x) + u(x)v′(x)
= cos(x)·cos(x) + sin(x)·(−sin(x))
= cos2(x)−sin2(x)
= cos(2x)
Step 6: Therefore, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos(2x).
Question 7
Question
Find the derivative of f(x) = sin(2x) cos(3x).
Solution
Step 1: Apply the product rule to differentiate f(x).
f′(x) = sin(2x)·d
dx (cos(3x))+d
dx (sin(2x)) ·cos(3x)
Step 2: Compute the derivative of cos(3x) and sin(2x).
d
dx (cos(3x)) = −3 sin(3x)
d
dx (sin(2x)) = 2 cos(2x)
Step 3: Substitute the derivatives back into f′(x).
f′(x) = (sin(2x)(−3 sin(3x))) + (2 cos(2x)·cos(3x))
=−3 sin(2x) sin(3x) + 2 cos(2x) cos(3x)
Therefore, the derivative of f(x) = sin(2x) cos(3x) is f′(x) = −3 sin(2x) sin(3x)+
2 cos(2x) cos(3x).
Question 8
Question
Find the derivative of the function f(x) = csc(2x) + cot(3x).
5
Solution
Step 1: Recall the derivatives of the trigonometric functions. The derivatives of
the six trigonometric functions are as follows:
d
dx (sin(x)) = cos(x),d
dx (cos(x)) = −sin(x),
d
dx (tan(x)) = sec2(x),d
dx (csc(x)) = −csc(x) cot(x),
d
dx (sec(x)) = sec(x) tan(x),d
dx (cot(x)) = −csc2(x).
Step 2: Find the derivative of f(x) = csc(2x) + cot(3x) using the above
derivatives.
d
dx (csc(2x)) = −csc(2x) cot(2x)·2 = −2 csc(2x) cot(2x)
d
dx (cot(3x)) = −csc2(3x)·3 = −3 csc2(3x)
Step 3: Combine the derivatives to find the derivative of f(x).
f′(x) = −2 csc(2x) cot(2x)−3 csc2(3x)
Therefore, the derivative of f(x) = csc(2x)+cot(3x) is f′(x) = −2 csc(2x) cot(2x)−3 csc2(3x) .
Question 9
Question
Find the derivative of the function f(x) = cos2(x) sin(2x).
Solution
Step 1: Use the product rule to differentiate f(x) = cos2(x) sin(2x). Step 2:
Let u= cos2(x) and v= sin(2x). Step 3: Compute u′=−2 cos(x) sin(x) and
v′= 2 cos(2x). Step 4: Apply the product rule f′(x) = u′v+uv′. Step 5:
Substitute the values of u,u′,vand v′into the product rule formula. Step 6:
Simplify the expression to find the final derivative of f(x).
Question 10
Question
Find the derivative of the following function:
f(x) = sin(x) + cos(x)
sin(x)−cos(x)
6
Solution
To find the derivative of the given function, we need to apply the quotient rule.
Let u(x) = sin(x) + cos(x) and v(x) = sin(x)−cos(x).
Step 1: Determine u′(x) and v′(x).
For u(x) = sin(x) + cos(x):
u′(x) = cos(x)−sin(x)
For v(x) = sin(x)−cos(x):
v′(x) = cos(x) + sin(x)
Step 2: Apply the quotient rule to find f′(x).
f′(x) = u′(x)v(x)−v′(x)u(x)
(v(x))2
Step 3: Substitute u′(x), v′(x), u(x), and v(x) into the quotient rule for-
mula.
f′(x) = (cos(x)−sin(x))(sin(x)−cos(x)) −(cos(x) + sin(x))(sin(x) + cos(x))
(sin(x)−cos(x))2
=cos(x) sin(x)−cos2(x)−sin2(x) + sin(x) cos(x)−cos(x) sin(x)−sin2(x)−cos(x) sin(x)−sin(x) cos(x)
(sin(x)−cos(x))2
=−2 cos2(x)−2 sin2(x)
(sin(x)−cos(x))2
=−2(cos2(x) + sin2(x))
(sin(x)−cos(x))2
=−2
(sin(x)−cos(x))2
Therefore, the derivative of the function f(x) = sin(x)+cos(x)
sin(x)−cos(x)is f′(x) =
−2
(sin(x)−cos(x))2.
Question 11
Question
Find the derivative of y= sin(2x) + cos(3x).
Solution
Step 1: Use the chain rule to find the derivative of sin(2x).
d
dx [sin(2x)] = cos(2x)·d
dx (2x)
7
Step 2: Simplify the derivative of sin(2x).
d
dx [sin(2x)] = 2 cos(2x)
Step 3: Use the chain rule to find the derivative of cos(3x).
d
dx [cos(3x)] = −sin(3x)·d
dx (3x)
Step 4: Simplify the derivative of cos(3x).
d
dx [cos(3x)] = −3 sin(3x)
Step 5: Combine the derivatives of sin(2x) and cos(3x) to find the derivative
of y.
y′= 2 cos(2x)−3 sin(3x)
Therefore, the derivative of y= sin(2x)+cos(3x) is y′= 2 cos(2x)−3 sin(3x).
Question 12
Question
Differentiate the function f(x) = tan(x)
cos(x).
Solution
Step 1: Rewrite the function using trigonometric identities. Step 2: Differentiate
using the quotient rule.
Step 1: We rewrite the function f(x) = tan(x)
cos(x)using trigonometric identities:
f(x) = sin(x)
cos(x)·1
cos(x)
f(x) = sin(x)
cos2(x)
Step 2: Now we differentiate f(x) using the quotient rule. If f(x) = g(x)
h(x),
then f′(x) = g′(x)h(x)−g(x)h′(x)
(h(x))2. Applying the quotient rule to f(x) = sin(x)
cos2(x),
we have:
f′(x) = cos2(x) cos(x)−sin(x)(−2 cos(x) sin(x))
cos4(x)
f′(x) = cos3(x) + 2 sin2(x) cos(x)
cos4(x)
f′(x) = cos(x)(1 + 2 sin2(x))
cos4(x)
8
f′(x) = cos(x)(1 + 2 sin2(x))
cos4(x)
f′(x) = cos(x)(1 + 2(1 −cos2(x)))
cos4(x)
f′(x) = cos(x)(1 + 2 −2 cos2(x))
cos4(x)
f′(x) = cos(x)(3 −2 cos2(x))
cos4(x)
f′(x) = 3 cos(x)−2 cos3(x)
cos4(x)
f′(x) = 3−2 cos2(x)
cos3(x)
Therefore, the derivative of f(x) = tan(x)
cos(x)is 3−2 cos2(x)
cos3(x).
Question 13
Question
Find the derivative of f(x) = sin(2x) cos(3x).
Solution
Step 1: Apply the product rule, which states that if f(x) = u(x)v(x), then
f′(x) = u′v+uv′.
Step 2: Let u(x) = sin(2x) and v(x) = cos(3x). Then, we need to find u′(x)
and v′(x).
Step 3: Find u′(x) by applying the chain rule. Since the derivative of sin(u)
is cos(u)u′, we have u′(x) = cos(2x)·2.
Step 4: Find v′(x) by applying the chain rule. Since the derivative of cos(u)
is −sin(u)u′, we have v′(x) = −sin(3x)·3.
Step 5: Now, apply the product rule to find f′(x):
f′(x) = u′v+uv′= (cos(2x)·2)(cos(3x)) + (sin(2x))(−sin(3x)·3)
Step 6: Simplify the expression to get the final answer for f′(x).
f′(x) = 2 cos(2x) cos(3x)−3 sin(2x) sin(3x)
Question 14
Question
Find the derivative of the function f(x) = sin(2x) cos(3x).
9
Solution
To find the derivative of f(x) = sin(2x) cos(3x), we will use the product rule.
Step 1: Apply the product rule. Let u= sin(2x) and v= cos(3x). Then,
using the product rule f′(x) = u′v+uv′, we have:
f′(x) = (2 cos(2x))(cos(3x)) + (sin(2x))(−3 sin(3x))
Step 2: Simplify the derivative. Substituting u= sin(2x), u′= 2 cos(2x),
v= cos(3x), and v′=−3 sin(3x) into the formula:
f′(x) = 2 cos(2x) cos(3x)−3 sin(2x) sin(3x)
Thus, the derivative of the function f(x) = sin(2x) cos(3x) is f′(x) =
2 cos(2x) cos(3x)−3 sin(2x) sin(3x).
Question 15
Question
Find the derivative of y= cos2(3x) sin(2x) with respect to x.
Solution
Step 1: Apply the product rule to differentiate the function. Let u= cos2(3x)
and v= sin(2x). Then, y=uv, and the derivative of ycan be found using the
formula dy
dx =udv
dx +vdu
dx .
Step 2: Find du
dx and dv
dx .
du
dx =d
dx (cos2(3x)) = 2 cos(3x)(−3 sin(3x)) = −6 cos(3x) sin(3x)
dv
dx =d
dx (sin(2x)) = 2 cos(2x)
Step 3: Substitute u,v,du
dx , and dv
dx into the product rule formula.
dy
dx = (cos2(3x))(2 cos(2x)) + (sin(2x))(−6 cos(3x) sin(3x))
Step 4: Simplify the expression.
dy
dx = 2 cos3(3x) cos(2x)−6 sin(2x) cos(3x) sin(3x)
Therefore, the derivative of y= cos2(3x) sin(2x) with respect to xis 2 cos3(3x) cos(2x)−
6 sin(2x) cos(3x) sin(3x).
10
Question 16
Question
Find the derivative of the function f(x) = sin(2x) + cos(3x).
Solution
Step 1: We will differentiate each term of the function separately using the rules
for differentiating trigonometric functions. Step 2: Recall that the derivative
of sin(u) is cos(u) multiplied by the derivative of u. Similarly, the derivative of
cos(v) is −sin(v) multiplied by the derivative of v.
Therefore, the derivative of f(x) can be found as follows:
f′(x) = d
dx [sin(2x)] + d
dx [cos(3x)]
Step 3: Applying the chain rule, we get:
f′(x) = cos(2x)·d
dx (2x)−sin(3x)·d
dx (3x)
Step 4: Simplifying, we have:
f′(x) = 2 cos(2x)−3 sin(3x)
Step 5: Therefore, the derivative of the function f(x) = sin(2x) + cos(3x) is
f′(x) = 2 cos(2x)−3 sin(3x).
Question 17
Question
Differentiate the function f(x) = sinx2+ cos(2x) with respect to x.
Solution
Step 1: Apply the chain rule to differentiate sinx2.
d
dx (sinx2) = cosx2·d
dx (x2).
Step 2: Simplify the derivative.
d
dx (sinx2) = cosx2·2x= 2xcosx2.
Step 3: Apply the chain rule to differentiate cos(2x).
d
dx (cos(2x)) = −sin(2x)·d
dx (2x).
11
Step 4: Simplify the derivative.
d
dx (cos(2x)) = −sin(2x)·2 = −2 sin(2x).
Step 5: Combine the derivatives to find f′(x).
f′(x) = 2xcosx2−2 sin(2x).
Therefore, the derivative of f(x) = sinx2+ cos(2x) with respect to xis
f′(x) = 2xcosx2−2 sin(2x) .
Question 18
Question
Find the derivative of the function f(x) = cos(2x)·sin(3x).
Solution
Step 1: Apply the product rule to differentiate the product of cos(2x) and
sin(3x). Step 2: Let u= cos(2x) and v= sin(3x). Step 3: Find u′and v′by
differentiating uand vwith respect to x. Step 4: Calculate f′(x) by applying
the product rule formula f′(x) = u′v+uv′. Step 5: Simplify the derivative
expression to get the final answer.
Step 1: Apply the product rule:
f′(x) = (cos(2x))′·sin(3x) + cos(2x)·(sin(3x)′)
Step 2: Let u= cos(2x) and v= sin(3x).
Step 3: Find u′and v′:
u′=−2 sin(2x), v′= 3 cos(3x)
Step 4: Substitute u′and v′into the product rule formula:
f′(x)=(−2 sin(2x)·sin(3x)) + (cos(2x)·3 cos(3x))
Step 5: Simplify the expression:
f′(x) = −2 sin(2x) sin(3x) + 3 cos(2x) cos(3x)
Therefore, the derivative of the function f(x) = cos(2x)·sin(3x) is −2 sin(2x) sin(3x)+
3 cos(2x) cos(3x).
Question 19
Question
Find the derivative of the function f(x) = sin2(x) cos(x).
12
Solution
Step 1: We will use the product rule to find the derivative of the given function
f(x). Step 2: Recall that the product rule states: (uv)′=u′v+uv′, where uand
vare differentiable functions of x. Step 3: Let u(x) = sin2(x) and v(x) = cos(x).
Step 4: Find the derivatives of u(x) and v(x):
u′(x) = 2 sin(x) cos(x)
v′(x) = −sin(x)
Step 5: Apply the product rule to find f′(x):
f′(x) = u′v+uv′
= (2 sin(x) cos(x))(cos(x)) + (sin2(x))(−sin(x))
= 2 sin(x) cos2(x)−sin3(x)
Step 6: Therefore, the derivative of f(x) = sin2(x) cos(x) is f′(x) = 2 sin(x) cos2(x)−sin3(x) .
Question 20
Question
Let f(x) = cos2(2x). Find f′(x).
Solution
Step 1: Apply the chain rule.
f′(x) = d
dx (cos2(2x))
= 2 cos(2x)·−sin(2x)·d
dx (2x)
Step 2: Simplify by finding d
dx (2x).
= 2 cos(2x)·(−sin(2x)·2)
Step 3: Simplify the expression.
=−4 cos(2x) sin(2x)
Step 4: Use the double-angle formula sin(2x) = 2 sin(x) cos(x).
=−4 cos(2x)·2 sin(x) cos(x)
Step 5: Apply the double-angle formula cos(2x) = 2 cos2(x)−1.
=−4(2 cos2(x)−1) ·2 sin(x) cos(x)
13
Step 6: Simplify and expand the expression.
=−8 cos2(x) sin(x) cos(x) + 8 sin(x) cos(x)
Step 7: Use the double-angle formula sin(2x) = 2 sin(x) cos(x) to simplify.
=−8 cos2(x) sin(x) cos(x) + 8 sin(x) cos(x)
Step 8: Finally, simplify the expression.
=−8 cos2(x) sin(x) + 8 sin(x) cos(x)
Question 21
Question
Find the derivative of f(x) = sin2(x) cos(2x).
Solution
To find the derivative of f(x), we will use the product rule and chain rule of
differentiation.
Step 1: Apply the product rule. Let u= sin2(x) and v= cos(2x). Then,
f′(x) = u′v+uv′
Step 2: Find u′and v′.
Derivative of u= sin2(x):
u′= 2 sin(x) cos(x)
Derivative of v= cos(2x):
v′=−2 sin(2x)
Step 3: Substitute u′and v′back into the product rule formula.
f′(x) = (2 sin(x) cos(x))(cos(2x)) + (sin2(x))(−2 sin(2x))
Step 4: Simplify the expression.
f′(x) = 2 sin(x) cos(x) cos(2x)−2 sin2(x) sin(2x)
Step 5: Use trigonometric identities to further simplify the expression. For
example, use the double angle identities:
sin(2x) = 2 sin(x) cos(x)
cos(2x) = cos2(x)−sin2(x)
Step 6: Substitute the identities into the derivative expression and simplify
further if needed.
14
Question 22
Question
Find the derivative of the function f(x) = cos2(3x) + sin2(2x).
Solution
Step 1: Recall the trigonometric identity cos2(x) + sin2(x) = 1 for any angle x.
Step 2: Rewrite the function f(x) using this identity: f(x) = 1.
Step 3: The derivative of a constant function is 0.
Step 4: Therefore, the derivative of f(x) = cos2(3x) + sin2(2x) is 0 .
Question 23
Question
Find the derivative of the function f(x) = sin2(3x) + cos2(5x).
Solution
Step 1: Use the identity sin2(x) + cos2(x) = 1 to simplify the function. Step
2: Differentiate the simplified function using the chain rule for trigonometric
functions. Step 3: Rewrite the derivative in terms of cosine functions if needed.
Step 1: First, let’s simplify the given function f(x) using the trigonometric
identity sin2(x) + cos2(x) = 1:
f(x) = sin2(3x) + cos2(5x)
= 1 + 1
= 2
So, f(x) = 2.
Step 2: Now, let’s find the derivative of the function f(x) = 2 with respect
to x:d
dx f(x) = d
dx 2
= 0
Therefore, the derivative of f(x) = sin2(3x) + cos2(5x) is 0.
Step 3: The derivative of a constant function is 0, so the derivative of f(x)
is indeed 0.
Question 24
Question
Find the derivative of the function f(x) = sin(3x) cos(2x).
15
Solution
To find the derivative of f(x) = sin(3x) cos(2x), we will use the product rule,
which states that if h(x) = g(x)·k(x), then h′(x) = g′(x)·k(x) + g(x)·k′(x).
Step 1: Identify g(x) and k(x) in f(x) = sin(3x) cos(2x).
Here, g(x) = sin(3x) and k(x) = cos(2x).
Step 2: Find g′(x) and k′(x).
Using the chain rule, g′(x) = 3 cos(3x) and k′(x) = −2 sin(2x).
Step 3: Apply the product rule.
The derivative of f(x) is:
f′(x) = g′(x)·k(x) + g(x)·k′(x)
= 3 cos(3x)·cos(2x) + sin(3x)·(−2 sin(2x))
= 3 cos(3x) cos(2x)−2 sin(3x) sin(2x).
Therefore, the derivative of f(x) = sin(3x) cos(2x) is f′(x) = 3 cos(3x) cos(2x)−
2 sin(3x) sin(2x).
Question 25
Question
Find the derivative of the function f(x) = sin(x) cos(x).
Solution
Step 1: Apply the product rule.
Let u= sin(x) and v= cos(x). Then, the product rule states that the
derivative of u·vis given by:
(u·v)′=u′v+uv′
Step 2: Find u′and v′.
Since d
dx (sin(x)) = cos(x) and d
dx (cos(x)) = −sin(x), we get:
u′= cos(x)
v′=−sin(x)
Step 3: Substitute u′and v′into the product rule formula.
So, the derivative of f(x) = sin(x) cos(x) can be found using the product
rule as:
f′(x) = (sin(x)·cos(x))′= (cos(x)·cos(x) + sin(x)· − sin(x))
f′(x) = cos2(x)−sin2(x)
Therefore, the derivative of the function f(x) = sin(x) cos(x) is f′(x) =
cos2(x)−sin2(x).
16
Question 26
Question
Find the derivative of the function f(x) = sin(2x) cos(3x).
Solution
Step 1: Apply the product rule to find the derivative of the function f(x) =
sin(2x) cos(3x). Step 2: Let u= sin(2x) and v= cos(3x). Step 3: Find u′
and v′. Step 4: Apply the product rule: f′(x) = u′v+uv′. Step 5: Substitute
back u,v,u′, and v′into the formula. Step 6: Simplify the expression for the
derivative.
Question 27
Question
Find the derivative of the function f(x) = 3 sin(2x) + cos(x).
Solution
To find the derivative of f(x) = 3 sin(2x)+cos(x), we will use the differentiation
rules for trigonometric functions.
Step 1: Find f′(x) by applying the derivative rules.
Recall the derivative rules: - d
dx [sin(ax)] = acos(ax) - d
dx [cos(ax)] = −asin(ax)
Using these rules, we find:
f′(x) = d
dx [3 sin(2x)] + d
dx [cos(x)]
= 3(2 cos(2x)) −sin(x)
= 6 cos(2x)−sin(x).
Therefore, the derivative of f(x) = 3 sin(2x)+cos(x) is f′(x) = 6 cos(2x)−sin(x) .
Question 28
Question
Find the derivative of the function f(x) = sin2(x) cos2(x).
17
Solution
Step 1: Use the product rule to differentiate f(x) = sin2(x) cos2(x).
f′(x) = d
dx (sin2(x)) ·cos2(x) + sin2(x)·d
dx (cos2(x))
= 2 sin(x)·cos(x)·cos2(x) + sin2(x)·2 cos(x)·(−sin(x))
= 2 sin(x)·cos(x)·cos2(x)−2 sin(x)·cos(x)·sin2(x)
Step 2: Factor out a common factor of 2 sin(x)·cos(x).
f′(x) = 2 sin(x)·cos(x)·(cos2(x)−sin2(x))
= 2 sin(x)·cos(x)·(cos(2x))
Therefore, the derivative of the function f(x) = sin2(x) cos2(x) is f′(x) =
2 sin(x)·cos(x)·cos(2x).
Question 29
Question
Find the derivative of the function f(x) = sec(x) + tan(x).
Solution
Step 1: We will differentiate each term separately using the rules of differen-
tiation. Step 2: Recall that d
dx (sec(x)) = sec(x) tan(x). Step 3: Similarly,
d
dx (tan(x)) = sec2(x). Step 4: Therefore, the derivative of f(x) = sec(x)+tan(x)
is d
dx (f(x)) = d
dx (sec(x)) + d
dx (tan(x))
Step 5: Substituting the derivatives we found earlier, we get
d
dx (f(x)) = sec(x) tan(x) + sec2(x)
Step 6: Thus, the derivative of the function f(x) = sec(x)+tan(x) is sec(x) tan(x)+
sec2(x).
Question 30
Question
Find the derivative of the function f(x) = sin(x) cos(x).
18
Solution
To find the derivative of f(x) = sin(x) cos(x), we will use the product rule,
which states that the derivative of a product of two functions is the first function
times the derivative of the second function, plus the second function times the
derivative of the first function.
Step 1: Identify the functions uand v. Let u(x) = sin(x) and v(x) = cos(x).
Step 2: Find u′(x) and v′(x). Using the derivatives of sine and cosine
functions, we have: u′(x) = cos(x) and v′(x) = −sin(x).
Step 3: Apply the product rule. The derivative of f(x) = sin(x) cos(x) is:
f′(x) = u′v+uv′
Substitute u′,v′,u, and vinto the formula:
f′(x) = cos(x) cos(x) + sin(x)(−sin(x))
Step 4: Simplify the expression. Simplify the expression to get the final
answer:
f′(x) = cos2(x)−sin2(x)
Therefore, the derivative of the function f(x) = sin(x) cos(x) is f′(x) =
cos2(x)−sin2(x).
Question 31
Question
Find the derivative of f(x) = sin2(2x) + cos(3x).
Solution
Step 1: Apply the chain rule to differentiate sin2(2x).
Let u= sin(2x) =⇒du
dx = 2 cos(2x)
⇒d
dx (sin2(2x)) = 2 sin(2x) cos(2x)
Step 2: Differentiate cos(3x) using the chain rule.
d
dx (cos(3x)) = −3 sin(3x)
Step 3: Combining the derivatives, we get the derivative of f(x).
d
dx (f(x)) = 2 sin(2x) cos(2x)−3 sin(3x)
Therefore, the derivative of f(x) = sin2(2x) + cos(3x) is 2 sin(2x) cos(2x)−
3 sin(3x).
19
Question 32
Question
Find the derivative of f(x) = sin(2x) cos(3x) with respect to x.
Solution
To find the derivative of f(x) = sin(2x) cos(3x) with respect to x, we will use the
product rule of differentiation: (uv)′=u′v+uv′, where uand vare functions
of x.
Step 1: Let u= sin(2x) and v= cos(3x). Then we have:
u′= 2 cos(2x) and v′=−3 sin(3x)
Step 2: Apply the product rule:
(f(x))′= (uv)′
=u′v+uv′
= (2 cos(2x))(cos(3x)) + (sin(2x))(−3 sin(3x))
= 2 cos(2x) cos(3x)−3 sin(2x) sin(3x)
Hence, the derivative of f(x) = sin(2x) cos(3x) with respect to xis 2 cos(2x) cos(3x)−
3 sin(2x) sin(3x).
Question 33
Question
Find the derivative of the function f(x) = cos(x)−2 sin(x)
tan(x).
Solution
To find the derivative of the given function, f(x) = cos(x)−2 sin(x)
tan(x), we will use
the quotient rule.
Step 1: Apply the quotient rule. Let u(x) = cos(x)−2 sin(x) and v(x) =
tan(x). The quotient rule states that the derivative of u(x)
v(x)is given by:
d
dx u(x)
v(x)=v(x)u′(x)−u(x)v′(x)
(v(x))2
Step 2: Calculate the derivatives of u(x) and v(x). We have: u′(x) =
−sin(x)−2 cos(x) (derivative of cos(x)−2 sin(x))
v′(x) = sec2(x) (derivative of tan(x))
20
Step 3: Apply the quotient rule formula. Substitute u′(x), v(x), u(x), and
v′(x) into the formula:
f′(x) = tan(x)(−sin(x)−2 cos(x)) −(cos(x)−2 sin(x)) sec2(x)
(tan(x))2
Step 4: Simplify the expression.
f′(x) = −sin(x) tan(x)−2 cos(x) tan(x)−cos(x) sec2(x) + 2 sin(x) sec2(x)
tan2(x)
Step 5: Further simplify the expression. Using trigonometric identities, we
can simplify the expression further. For example,
tan(x) = sin(x)
cos(x)and sec2(x) = 1 + tan2(x)
Step 6: Final answer. After simplification, the derivative of f(x) is:
f′(x) = −3 sin(x)−2 cos(x)−cos(x)−tan(x)
cos2(x)
Question 34
Question
Find the derivative of the function f(x) = sin2(x) cos(x).
Solution
Step 1: Apply the product rule to the function f(x) = sin2(x) cos(x). Step 2:
Let u= sin2(x) and v= cos(x). Step 3: Find u′and v′. Step 4: Differentiate u
with respect to x:u′= 2 sin(x) cos(x). Step 5: Differentiate vwith respect to
x:v′=−sin(x). Step 6: Apply the product rule: f′(x) = u′v+uv′. Step 7:
Substitute u′,v′,u, and vinto the product rule formula. Step 8: Simplify the
expression to find the derivative of f(x).
Question 35
Question
Find the derivative of y= sin(3x) + cos(2x) with respect to x.
Solution
Step 1: Use the fact that d
dx (sin(ax)) = acos(ax) and d
dx (cos(ax)) = −asin(ax)
for any constant a.
21
Solution
To find the derivative of f(x) = tan2(x)−sin(x) cos(x), we will differentiate
term by term using the rules of differentiation.
Step 1: Let’s find the derivative of tan2(x).
d
dx (tan2(x)) = 2 tan(x) sec2(x)
Step 2: Now, let’s find the derivative of sin(x) cos(x).
d
dx (sin(x) cos(x)) = cos2(x)−sin2(x)
Step 3: Putting it all together, the derivative of f(x) = tan2(x)−sin(x) cos(x)
is:
d
dx (f(x)) = 2 tan(x) sec2(x)−(cos2(x)−sin2(x))
= 2 tan(x) sec2(x)−cos2(x) + sin2(x)
Therefore, the derivative of f(x) = tan2(x)−sin(x) cos(x) is 2 tan(x) sec2(x)−
cos2(x) + sin2(x).
Question 3
Question
Find the derivative of y= sin(2x) cos(3x) with respect to x.
Solution
Step 1: Apply the product rule for differentiation, which states that the deriva-
tive of a product of two functions is the derivative of the first function times
the second function, plus the first function times the derivative of the sec-
ond function. Step 2: Let u= sin(2x) and v= cos(3x). Step 3: Find
the derivatives of uand vwith respect to x. Step 4: du
dx = 2 cos(2x) and
dv
dx =−3 sin(3x). Step 5: Apply the product rule: dy
dx =udv
dx +vdu
dx . Step
6: Substitute the values of u,v,du
dx , and dv
dx into the formula above. Step
7: dy
dx = sin(2x)·(−3 sin(3x)) + cos(3x)·2 cos(2x). Step 8: Simplify the ex-
pression in Step 7 by expanding and using trigonometric identities. Step 9:
dy
dx =−3 sin(2x) sin(3x) + 2 cos(3x) cos(2x). Step 10: Use the trigonometric
identity sin(A) sin(B) = 1
2[cos(A−B)−cos(A+B)] to simplify the expression
further. Step 11: dy
dx =−3
2[cos(2x+ 3x)−cos(2x−3x)] + 2 cos(3x) cos(2x).
Step 12: Simplify the angles in the cosine terms. Step 13: dy
dx =−3
2[cos(5x)−
cos(−x)] + 2 cos(3x) cos(2x). Step 14: Note that cos(−x) = cos(x). Step 15:
dy
dx =−3
2[cos(5x)−cos(x)] + 2 cos(3x) cos(2x). Therefore, the derivative of y=
sin(2x) cos(3x) with respect to xis dy
dx =−3
2[cos(5x)−cos(x)]+2 cos(3x) cos(2x).
2
Question 4
Question
Find the derivative of the function f(x) = sin(x) cos(x).
Solution
Step 1: Apply the product rule, which states that the derivative of a product
of two functions is the derivative of the first function times the second function,
plus the first function times the derivative of the second function. Let u= sin(x)
and v= cos(x). Then, we have
f(x) = u·v
Step 2: Calculate the derivatives of uand v.
Derivative of u= sin(x):
du
dx = cos(x)
Derivative of v= cos(x):
dv
dx =−sin(x)
Step 3: Apply the product rule to find the derivative of f(x).
f′(x) = u′·v+u·v′
Step 4: Substitute the derivatives of uand vinto the formula for f′(x).
f′(x) = cos(x)·cos(x) + sin(x)·(−sin(x))
Step 5: Simplify the expression.
f′(x) = cos2(x)−sin2(x)
Step 6: Recall the trigonometric identity cos2(x)−sin2(x) = cos(2x). There-
fore, the derivative of f(x) = sin(x) cos(x) is:
f′(x) = cos(2x)
Question 5
Question
Find the derivative of the function f(x) = sin2(x) cos(x).
3
Solution
Step 1: Use the product rule to differentiate the function f(x). Step 2: Apply
the chain rule to differentiate sin2(x). Step 3: Simplify the expression to find
the derivative of f(x).
Step 1: The product rule states that if u(x) and v(x) are differentiable
functions, then the derivative of their product is given by:
(uv)′=u′v+uv′
In this case, u(x) = sin2(x) and v(x) = cos(x). Therefore, the derivative of f(x)
is:
f′(x) = (sin2(x))′cos(x) + sin2(x)(cos(x))′
Step 2: To differentiate sin2(x), we need to apply the chain rule. Let
u= sin(x), then d
dx (sin2(x)) = d
dx (sin(u)2) = 2 sin(x) cos(x).
Step 3: Substitute the derivatives back into the expression for f′(x):
f′(x) = 2 sin(x) cos(x) cos(x) + sin2(x)(−sin(x))
Simplify the expression:
f′(x) = 2 sin(x) cos2(x)−sin3(x)
So, the derivative of the function f(x) = sin2(x) cos(x) is f′(x) = 2 sin(x) cos2(x)−
sin3(x).
Question 6
Question
Let f(x) = sin(x) cos(x). Find f′(x).
Solution
Step 1: We will use the product rule to differentiate f(x) = sin(x) cos(x).
Step 2: Recall that the product rule states that if f(x) = u(x)v(x), then
f′(x) = u′(x)v(x) + u(x)v′(x).
Step 3: Let u(x) = sin(x) and v(x) = cos(x).
Step 4: Find u′(x) and v′(x).
u′(x) = d
dx (sin(x))
= cos(x)
v′(x) = d
dx (cos(x))
=−sin(x)
4
Step 5: Apply the product rule to find f′(x).
f′(x) = u′(x)v(x) + u(x)v′(x)
= cos(x)·cos(x) + sin(x)·(−sin(x))
= cos2(x)−sin2(x)
= cos(2x)
Step 6: Therefore, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos(2x).
Question 7
Question
Find the derivative of f(x) = sin(2x) cos(3x).
Solution
Step 1: Apply the product rule to differentiate f(x).
f′(x) = sin(2x)·d
dx (cos(3x))+d
dx (sin(2x)) ·cos(3x)
Step 2: Compute the derivative of cos(3x) and sin(2x).
d
dx (cos(3x)) = −3 sin(3x)
d
dx (sin(2x)) = 2 cos(2x)
Step 3: Substitute the derivatives back into f′(x).
f′(x) = (sin(2x)(−3 sin(3x))) + (2 cos(2x)·cos(3x))
=−3 sin(2x) sin(3x) + 2 cos(2x) cos(3x)
Therefore, the derivative of f(x) = sin(2x) cos(3x) is f′(x) = −3 sin(2x) sin(3x)+
2 cos(2x) cos(3x).
Question 8
Question
Find the derivative of the function f(x) = csc(2x) + cot(3x).
5
Solution
Step 1: Recall the derivatives of the trigonometric functions. The derivatives of
the six trigonometric functions are as follows:
d
dx (sin(x)) = cos(x),d
dx (cos(x)) = −sin(x),
d
dx (tan(x)) = sec2(x),d
dx (csc(x)) = −csc(x) cot(x),
d
dx (sec(x)) = sec(x) tan(x),d
dx (cot(x)) = −csc2(x).
Step 2: Find the derivative of f(x) = csc(2x) + cot(3x) using the above
derivatives.
d
dx (csc(2x)) = −csc(2x) cot(2x)·2 = −2 csc(2x) cot(2x)
d
dx (cot(3x)) = −csc2(3x)·3 = −3 csc2(3x)
Step 3: Combine the derivatives to find the derivative of f(x).
f′(x) = −2 csc(2x) cot(2x)−3 csc2(3x)
Therefore, the derivative of f(x) = csc(2x)+cot(3x) is f′(x) = −2 csc(2x) cot(2x)−3 csc2(3x) .
Question 9
Question
Find the derivative of the function f(x) = cos2(x) sin(2x).
Solution
Step 1: Use the product rule to differentiate f(x) = cos2(x) sin(2x). Step 2:
Let u= cos2(x) and v= sin(2x). Step 3: Compute u′=−2 cos(x) sin(x) and
v′= 2 cos(2x). Step 4: Apply the product rule f′(x) = u′v+uv′. Step 5:
Substitute the values of u,u′,vand v′into the product rule formula. Step 6:
Simplify the expression to find the final derivative of f(x).
Question 10
Question
Find the derivative of the following function:
f(x) = sin(x) + cos(x)
sin(x)−cos(x)
6
Solution
To find the derivative of the given function, we need to apply the quotient rule.
Let u(x) = sin(x) + cos(x) and v(x) = sin(x)−cos(x).
Step 1: Determine u′(x) and v′(x).
For u(x) = sin(x) + cos(x):
u′(x) = cos(x)−sin(x)
For v(x) = sin(x)−cos(x):
v′(x) = cos(x) + sin(x)
Step 2: Apply the quotient rule to find f′(x).
f′(x) = u′(x)v(x)−v′(x)u(x)
(v(x))2
Step 3: Substitute u′(x), v′(x), u(x), and v(x) into the quotient rule for-
mula.
f′(x) = (cos(x)−sin(x))(sin(x)−cos(x)) −(cos(x) + sin(x))(sin(x) + cos(x))
(sin(x)−cos(x))2
=cos(x) sin(x)−cos2(x)−sin2(x) + sin(x) cos(x)−cos(x) sin(x)−sin2(x)−cos(x) sin(x)−sin(x) cos(x)
(sin(x)−cos(x))2
=−2 cos2(x)−2 sin2(x)
(sin(x)−cos(x))2
=−2(cos2(x) + sin2(x))
(sin(x)−cos(x))2
=−2
(sin(x)−cos(x))2
Therefore, the derivative of the function f(x) = sin(x)+cos(x)
sin(x)−cos(x)is f′(x) =
−2
(sin(x)−cos(x))2.
Question 11
Question
Find the derivative of y= sin(2x) + cos(3x).
Solution
Step 1: Use the chain rule to find the derivative of sin(2x).
d
dx [sin(2x)] = cos(2x)·d
dx (2x)
7
Step 2: Simplify the derivative of sin(2x).
d
dx [sin(2x)] = 2 cos(2x)
Step 3: Use the chain rule to find the derivative of cos(3x).
d
dx [cos(3x)] = −sin(3x)·d
dx (3x)
Step 4: Simplify the derivative of cos(3x).
d
dx [cos(3x)] = −3 sin(3x)
Step 5: Combine the derivatives of sin(2x) and cos(3x) to find the derivative
of y.
y′= 2 cos(2x)−3 sin(3x)
Therefore, the derivative of y= sin(2x)+cos(3x) is y′= 2 cos(2x)−3 sin(3x).
Question 12
Question
Differentiate the function f(x) = tan(x)
cos(x).
Solution
Step 1: Rewrite the function using trigonometric identities. Step 2: Differentiate
using the quotient rule.
Step 1: We rewrite the function f(x) = tan(x)
cos(x)using trigonometric identities:
f(x) = sin(x)
cos(x)·1
cos(x)
f(x) = sin(x)
cos2(x)
Step 2: Now we differentiate f(x) using the quotient rule. If f(x) = g(x)
h(x),
then f′(x) = g′(x)h(x)−g(x)h′(x)
(h(x))2. Applying the quotient rule to f(x) = sin(x)
cos2(x),
we have:
f′(x) = cos2(x) cos(x)−sin(x)(−2 cos(x) sin(x))
cos4(x)
f′(x) = cos3(x) + 2 sin2(x) cos(x)
cos4(x)
f′(x) = cos(x)(1 + 2 sin2(x))
cos4(x)
8
f′(x) = cos(x)(1 + 2 sin2(x))
cos4(x)
f′(x) = cos(x)(1 + 2(1 −cos2(x)))
cos4(x)
f′(x) = cos(x)(1 + 2 −2 cos2(x))
cos4(x)
f′(x) = cos(x)(3 −2 cos2(x))
cos4(x)
f′(x) = 3 cos(x)−2 cos3(x)
cos4(x)
f′(x) = 3−2 cos2(x)
cos3(x)
Therefore, the derivative of f(x) = tan(x)
cos(x)is 3−2 cos2(x)
cos3(x).
Question 13
Question
Find the derivative of f(x) = sin(2x) cos(3x).
Solution
Step 1: Apply the product rule, which states that if f(x) = u(x)v(x), then
f′(x) = u′v+uv′.
Step 2: Let u(x) = sin(2x) and v(x) = cos(3x). Then, we need to find u′(x)
and v′(x).
Step 3: Find u′(x) by applying the chain rule. Since the derivative of sin(u)
is cos(u)u′, we have u′(x) = cos(2x)·2.
Step 4: Find v′(x) by applying the chain rule. Since the derivative of cos(u)
is −sin(u)u′, we have v′(x) = −sin(3x)·3.
Step 5: Now, apply the product rule to find f′(x):
f′(x) = u′v+uv′= (cos(2x)·2)(cos(3x)) + (sin(2x))(−sin(3x)·3)
Step 6: Simplify the expression to get the final answer for f′(x).
f′(x) = 2 cos(2x) cos(3x)−3 sin(2x) sin(3x)
Question 14
Question
Find the derivative of the function f(x) = sin(2x) cos(3x).
9
Solution
To find the derivative of f(x) = sin(2x) cos(3x), we will use the product rule.
Step 1: Apply the product rule. Let u= sin(2x) and v= cos(3x). Then,
using the product rule f′(x) = u′v+uv′, we have:
f′(x) = (2 cos(2x))(cos(3x)) + (sin(2x))(−3 sin(3x))
Step 2: Simplify the derivative. Substituting u= sin(2x), u′= 2 cos(2x),
v= cos(3x), and v′=−3 sin(3x) into the formula:
f′(x) = 2 cos(2x) cos(3x)−3 sin(2x) sin(3x)
Thus, the derivative of the function f(x) = sin(2x) cos(3x) is f′(x) =
2 cos(2x) cos(3x)−3 sin(2x) sin(3x).
Question 15
Question
Find the derivative of y= cos2(3x) sin(2x) with respect to x.
Solution
Step 1: Apply the product rule to differentiate the function. Let u= cos2(3x)
and v= sin(2x). Then, y=uv, and the derivative of ycan be found using the
formula dy
dx =udv
dx +vdu
dx .
Step 2: Find du
dx and dv
dx .
du
dx =d
dx (cos2(3x)) = 2 cos(3x)(−3 sin(3x)) = −6 cos(3x) sin(3x)
dv
dx =d
dx (sin(2x)) = 2 cos(2x)
Step 3: Substitute u,v,du
dx , and dv
dx into the product rule formula.
dy
dx = (cos2(3x))(2 cos(2x)) + (sin(2x))(−6 cos(3x) sin(3x))
Step 4: Simplify the expression.
dy
dx = 2 cos3(3x) cos(2x)−6 sin(2x) cos(3x) sin(3x)
Therefore, the derivative of y= cos2(3x) sin(2x) with respect to xis 2 cos3(3x) cos(2x)−
6 sin(2x) cos(3x) sin(3x).
10
Question 16
Question
Find the derivative of the function f(x) = sin(2x) + cos(3x).
Solution
Step 1: We will differentiate each term of the function separately using the rules
for differentiating trigonometric functions. Step 2: Recall that the derivative
of sin(u) is cos(u) multiplied by the derivative of u. Similarly, the derivative of
cos(v) is −sin(v) multiplied by the derivative of v.
Therefore, the derivative of f(x) can be found as follows:
f′(x) = d
dx [sin(2x)] + d
dx [cos(3x)]
Step 3: Applying the chain rule, we get:
f′(x) = cos(2x)·d
dx (2x)−sin(3x)·d
dx (3x)
Step 4: Simplifying, we have:
f′(x) = 2 cos(2x)−3 sin(3x)
Step 5: Therefore, the derivative of the function f(x) = sin(2x) + cos(3x) is
f′(x) = 2 cos(2x)−3 sin(3x).
Question 17
Question
Differentiate the function f(x) = sinx2+ cos(2x) with respect to x.
Solution
Step 1: Apply the chain rule to differentiate sinx2.
d
dx (sinx2) = cosx2·d
dx (x2).
Step 2: Simplify the derivative.
d
dx (sinx2) = cosx2·2x= 2xcosx2.
Step 3: Apply the chain rule to differentiate cos(2x).
d
dx (cos(2x)) = −sin(2x)·d
dx (2x).
11
Step 4: Simplify the derivative.
d
dx (cos(2x)) = −sin(2x)·2 = −2 sin(2x).
Step 5: Combine the derivatives to find f′(x).
f′(x) = 2xcosx2−2 sin(2x).
Therefore, the derivative of f(x) = sinx2+ cos(2x) with respect to xis
f′(x) = 2xcosx2−2 sin(2x) .
Question 18
Question
Find the derivative of the function f(x) = cos(2x)·sin(3x).
Solution
Step 1: Apply the product rule to differentiate the product of cos(2x) and
sin(3x). Step 2: Let u= cos(2x) and v= sin(3x). Step 3: Find u′and v′by
differentiating uand vwith respect to x. Step 4: Calculate f′(x) by applying
the product rule formula f′(x) = u′v+uv′. Step 5: Simplify the derivative
expression to get the final answer.
Step 1: Apply the product rule:
f′(x) = (cos(2x))′·sin(3x) + cos(2x)·(sin(3x)′)
Step 2: Let u= cos(2x) and v= sin(3x).
Step 3: Find u′and v′:
u′=−2 sin(2x), v′= 3 cos(3x)
Step 4: Substitute u′and v′into the product rule formula:
f′(x)=(−2 sin(2x)·sin(3x)) + (cos(2x)·3 cos(3x))
Step 5: Simplify the expression:
f′(x) = −2 sin(2x) sin(3x) + 3 cos(2x) cos(3x)
Therefore, the derivative of the function f(x) = cos(2x)·sin(3x) is −2 sin(2x) sin(3x)+
3 cos(2x) cos(3x).
Question 19
Question
Find the derivative of the function f(x) = sin2(x) cos(x).
12
Solution
Step 1: We will use the product rule to find the derivative of the given function
f(x). Step 2: Recall that the product rule states: (uv)′=u′v+uv′, where uand
vare differentiable functions of x. Step 3: Let u(x) = sin2(x) and v(x) = cos(x).
Step 4: Find the derivatives of u(x) and v(x):
u′(x) = 2 sin(x) cos(x)
v′(x) = −sin(x)
Step 5: Apply the product rule to find f′(x):
f′(x) = u′v+uv′
= (2 sin(x) cos(x))(cos(x)) + (sin2(x))(−sin(x))
= 2 sin(x) cos2(x)−sin3(x)
Step 6: Therefore, the derivative of f(x) = sin2(x) cos(x) is f′(x) = 2 sin(x) cos2(x)−sin3(x) .
Question 20
Question
Let f(x) = cos2(2x). Find f′(x).
Solution
Step 1: Apply the chain rule.
f′(x) = d
dx (cos2(2x))
= 2 cos(2x)·−sin(2x)·d
dx (2x)
Step 2: Simplify by finding d
dx (2x).
= 2 cos(2x)·(−sin(2x)·2)
Step 3: Simplify the expression.
=−4 cos(2x) sin(2x)
Step 4: Use the double-angle formula sin(2x) = 2 sin(x) cos(x).
=−4 cos(2x)·2 sin(x) cos(x)
Step 5: Apply the double-angle formula cos(2x) = 2 cos2(x)−1.
=−4(2 cos2(x)−1) ·2 sin(x) cos(x)
13
Step 6: Simplify and expand the expression.
=−8 cos2(x) sin(x) cos(x) + 8 sin(x) cos(x)
Step 7: Use the double-angle formula sin(2x) = 2 sin(x) cos(x) to simplify.
=−8 cos2(x) sin(x) cos(x) + 8 sin(x) cos(x)
Step 8: Finally, simplify the expression.
=−8 cos2(x) sin(x) + 8 sin(x) cos(x)
Question 21
Question
Find the derivative of f(x) = sin2(x) cos(2x).
Solution
To find the derivative of f(x), we will use the product rule and chain rule of
differentiation.
Step 1: Apply the product rule. Let u= sin2(x) and v= cos(2x). Then,
f′(x) = u′v+uv′
Step 2: Find u′and v′.
Derivative of u= sin2(x):
u′= 2 sin(x) cos(x)
Derivative of v= cos(2x):
v′=−2 sin(2x)
Step 3: Substitute u′and v′back into the product rule formula.
f′(x) = (2 sin(x) cos(x))(cos(2x)) + (sin2(x))(−2 sin(2x))
Step 4: Simplify the expression.
f′(x) = 2 sin(x) cos(x) cos(2x)−2 sin2(x) sin(2x)
Step 5: Use trigonometric identities to further simplify the expression. For
example, use the double angle identities:
sin(2x) = 2 sin(x) cos(x)
cos(2x) = cos2(x)−sin2(x)
Step 6: Substitute the identities into the derivative expression and simplify
further if needed.
14
Question 22
Question
Find the derivative of the function f(x) = cos2(3x) + sin2(2x).
Solution
Step 1: Recall the trigonometric identity cos2(x) + sin2(x) = 1 for any angle x.
Step 2: Rewrite the function f(x) using this identity: f(x) = 1.
Step 3: The derivative of a constant function is 0.
Step 4: Therefore, the derivative of f(x) = cos2(3x) + sin2(2x) is 0 .
Question 23
Question
Find the derivative of the function f(x) = sin2(3x) + cos2(5x).
Solution
Step 1: Use the identity sin2(x) + cos2(x) = 1 to simplify the function. Step
2: Differentiate the simplified function using the chain rule for trigonometric
functions. Step 3: Rewrite the derivative in terms of cosine functions if needed.
Step 1: First, let’s simplify the given function f(x) using the trigonometric
identity sin2(x) + cos2(x) = 1:
f(x) = sin2(3x) + cos2(5x)
= 1 + 1
= 2
So, f(x) = 2.
Step 2: Now, let’s find the derivative of the function f(x) = 2 with respect
to x:d
dx f(x) = d
dx 2
= 0
Therefore, the derivative of f(x) = sin2(3x) + cos2(5x) is 0.
Step 3: The derivative of a constant function is 0, so the derivative of f(x)
is indeed 0.
Question 24
Question
Find the derivative of the function f(x) = sin(3x) cos(2x).
15
Solution
To find the derivative of f(x) = sin(3x) cos(2x), we will use the product rule,
which states that if h(x) = g(x)·k(x), then h′(x) = g′(x)·k(x) + g(x)·k′(x).
Step 1: Identify g(x) and k(x) in f(x) = sin(3x) cos(2x).
Here, g(x) = sin(3x) and k(x) = cos(2x).
Step 2: Find g′(x) and k′(x).
Using the chain rule, g′(x) = 3 cos(3x) and k′(x) = −2 sin(2x).
Step 3: Apply the product rule.
The derivative of f(x) is:
f′(x) = g′(x)·k(x) + g(x)·k′(x)
= 3 cos(3x)·cos(2x) + sin(3x)·(−2 sin(2x))
= 3 cos(3x) cos(2x)−2 sin(3x) sin(2x).
Therefore, the derivative of f(x) = sin(3x) cos(2x) is f′(x) = 3 cos(3x) cos(2x)−
2 sin(3x) sin(2x).
Question 25
Question
Find the derivative of the function f(x) = sin(x) cos(x).
Solution
Step 1: Apply the product rule.
Let u= sin(x) and v= cos(x). Then, the product rule states that the
derivative of u·vis given by:
(u·v)′=u′v+uv′
Step 2: Find u′and v′.
Since d
dx (sin(x)) = cos(x) and d
dx (cos(x)) = −sin(x), we get:
u′= cos(x)
v′=−sin(x)
Step 3: Substitute u′and v′into the product rule formula.
So, the derivative of f(x) = sin(x) cos(x) can be found using the product
rule as:
f′(x) = (sin(x)·cos(x))′= (cos(x)·cos(x) + sin(x)· − sin(x))
f′(x) = cos2(x)−sin2(x)
Therefore, the derivative of the function f(x) = sin(x) cos(x) is f′(x) =
cos2(x)−sin2(x).
16
Question 26
Question
Find the derivative of the function f(x) = sin(2x) cos(3x).
Solution
Step 1: Apply the product rule to find the derivative of the function f(x) =
sin(2x) cos(3x). Step 2: Let u= sin(2x) and v= cos(3x). Step 3: Find u′
and v′. Step 4: Apply the product rule: f′(x) = u′v+uv′. Step 5: Substitute
back u,v,u′, and v′into the formula. Step 6: Simplify the expression for the
derivative.
Question 27
Question
Find the derivative of the function f(x) = 3 sin(2x) + cos(x).
Solution
To find the derivative of f(x) = 3 sin(2x)+cos(x), we will use the differentiation
rules for trigonometric functions.
Step 1: Find f′(x) by applying the derivative rules.
Recall the derivative rules: - d
dx [sin(ax)] = acos(ax) - d
dx [cos(ax)] = −asin(ax)
Using these rules, we find:
f′(x) = d
dx [3 sin(2x)] + d
dx [cos(x)]
= 3(2 cos(2x)) −sin(x)
= 6 cos(2x)−sin(x).
Therefore, the derivative of f(x) = 3 sin(2x)+cos(x) is f′(x) = 6 cos(2x)−sin(x) .
Question 28
Question
Find the derivative of the function f(x) = sin2(x) cos2(x).
17
Solution
Step 1: Use the product rule to differentiate f(x) = sin2(x) cos2(x).
f′(x) = d
dx (sin2(x)) ·cos2(x) + sin2(x)·d
dx (cos2(x))
= 2 sin(x)·cos(x)·cos2(x) + sin2(x)·2 cos(x)·(−sin(x))
= 2 sin(x)·cos(x)·cos2(x)−2 sin(x)·cos(x)·sin2(x)
Step 2: Factor out a common factor of 2 sin(x)·cos(x).
f′(x) = 2 sin(x)·cos(x)·(cos2(x)−sin2(x))
= 2 sin(x)·cos(x)·(cos(2x))
Therefore, the derivative of the function f(x) = sin2(x) cos2(x) is f′(x) =
2 sin(x)·cos(x)·cos(2x).
Question 29
Question
Find the derivative of the function f(x) = sec(x) + tan(x).
Solution
Step 1: We will differentiate each term separately using the rules of differen-
tiation. Step 2: Recall that d
dx (sec(x)) = sec(x) tan(x). Step 3: Similarly,
d
dx (tan(x)) = sec2(x). Step 4: Therefore, the derivative of f(x) = sec(x)+tan(x)
is d
dx (f(x)) = d
dx (sec(x)) + d
dx (tan(x))
Step 5: Substituting the derivatives we found earlier, we get
d
dx (f(x)) = sec(x) tan(x) + sec2(x)
Step 6: Thus, the derivative of the function f(x) = sec(x)+tan(x) is sec(x) tan(x)+
sec2(x).
Question 30
Question
Find the derivative of the function f(x) = sin(x) cos(x).
18
Solution
To find the derivative of f(x) = sin(x) cos(x), we will use the product rule,
which states that the derivative of a product of two functions is the first function
times the derivative of the second function, plus the second function times the
derivative of the first function.
Step 1: Identify the functions uand v. Let u(x) = sin(x) and v(x) = cos(x).
Step 2: Find u′(x) and v′(x). Using the derivatives of sine and cosine
functions, we have: u′(x) = cos(x) and v′(x) = −sin(x).
Step 3: Apply the product rule. The derivative of f(x) = sin(x) cos(x) is:
f′(x) = u′v+uv′
Substitute u′,v′,u, and vinto the formula:
f′(x) = cos(x) cos(x) + sin(x)(−sin(x))
Step 4: Simplify the expression. Simplify the expression to get the final
answer:
f′(x) = cos2(x)−sin2(x)
Therefore, the derivative of the function f(x) = sin(x) cos(x) is f′(x) =
cos2(x)−sin2(x).
Question 31
Question
Find the derivative of f(x) = sin2(2x) + cos(3x).
Solution
Step 1: Apply the chain rule to differentiate sin2(2x).
Let u= sin(2x) =⇒du
dx = 2 cos(2x)
⇒d
dx (sin2(2x)) = 2 sin(2x) cos(2x)
Step 2: Differentiate cos(3x) using the chain rule.
d
dx (cos(3x)) = −3 sin(3x)
Step 3: Combining the derivatives, we get the derivative of f(x).
d
dx (f(x)) = 2 sin(2x) cos(2x)−3 sin(3x)
Therefore, the derivative of f(x) = sin2(2x) + cos(3x) is 2 sin(2x) cos(2x)−
3 sin(3x).
19
Question 32
Question
Find the derivative of f(x) = sin(2x) cos(3x) with respect to x.
Solution
To find the derivative of f(x) = sin(2x) cos(3x) with respect to x, we will use the
product rule of differentiation: (uv)′=u′v+uv′, where uand vare functions
of x.
Step 1: Let u= sin(2x) and v= cos(3x). Then we have:
u′= 2 cos(2x) and v′=−3 sin(3x)
Step 2: Apply the product rule:
(f(x))′= (uv)′
=u′v+uv′
= (2 cos(2x))(cos(3x)) + (sin(2x))(−3 sin(3x))
= 2 cos(2x) cos(3x)−3 sin(2x) sin(3x)
Hence, the derivative of f(x) = sin(2x) cos(3x) with respect to xis 2 cos(2x) cos(3x)−
3 sin(2x) sin(3x).
Question 33
Question
Find the derivative of the function f(x) = cos(x)−2 sin(x)
tan(x).
Solution
To find the derivative of the given function, f(x) = cos(x)−2 sin(x)
tan(x), we will use
the quotient rule.
Step 1: Apply the quotient rule. Let u(x) = cos(x)−2 sin(x) and v(x) =
tan(x). The quotient rule states that the derivative of u(x)
v(x)is given by:
d
dx u(x)
v(x)=v(x)u′(x)−u(x)v′(x)
(v(x))2
Step 2: Calculate the derivatives of u(x) and v(x). We have: u′(x) =
−sin(x)−2 cos(x) (derivative of cos(x)−2 sin(x))
v′(x) = sec2(x) (derivative of tan(x))
20
Step 3: Apply the quotient rule formula. Substitute u′(x), v(x), u(x), and
v′(x) into the formula:
f′(x) = tan(x)(−sin(x)−2 cos(x)) −(cos(x)−2 sin(x)) sec2(x)
(tan(x))2
Step 4: Simplify the expression.
f′(x) = −sin(x) tan(x)−2 cos(x) tan(x)−cos(x) sec2(x) + 2 sin(x) sec2(x)
tan2(x)
Step 5: Further simplify the expression. Using trigonometric identities, we
can simplify the expression further. For example,
tan(x) = sin(x)
cos(x)and sec2(x) = 1 + tan2(x)
Step 6: Final answer. After simplification, the derivative of f(x) is:
f′(x) = −3 sin(x)−2 cos(x)−cos(x)−tan(x)
cos2(x)
Question 34
Question
Find the derivative of the function f(x) = sin2(x) cos(x).
Solution
Step 1: Apply the product rule to the function f(x) = sin2(x) cos(x). Step 2:
Let u= sin2(x) and v= cos(x). Step 3: Find u′and v′. Step 4: Differentiate u
with respect to x:u′= 2 sin(x) cos(x). Step 5: Differentiate vwith respect to
x:v′=−sin(x). Step 6: Apply the product rule: f′(x) = u′v+uv′. Step 7:
Substitute u′,v′,u, and vinto the product rule formula. Step 8: Simplify the
expression to find the derivative of f(x).
Question 35
Question
Find the derivative of y= sin(3x) + cos(2x) with respect to x.
Solution
Step 1: Use the fact that d
dx (sin(ax)) = acos(ax) and d
dx (cos(ax)) = −asin(ax)
for any constant a.
21
Step 2: Find the derivative of each term separately.
d
dx (sin(3x)) = 3 cos(3x) and d
dx (cos(2x)) = −2 sin(2x)
Step 3: Combine the derivatives of the individual terms to find the derivative
of the entire expression y.
d
dx (y) = 3 cos(3x)−2 sin(2x)
Therefore, the derivative of y= sin(3x) + cos(2x) with respect to xis
3 cos(3x)−2 sin(2x).
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