MATH 108 - ELEMENTARY AND
INTERMEDIATE ALGEBRA -
Differentiation of Trigonometric
Functions
Question Bank - Set 1
Liberty University
Question 1
Question
Find the derivative of the function f(x) = sin(x) cos(x).
Solution
To find the derivative of f(x) = sin(x) cos(x), we can use the product rule.
Step 1: Apply the product rule, which states that the derivative of the
product of two functions uand vis given by
(uv)′=u′v+uv′.
Step 2: Let u= sin(x) and v= cos(x). Then, we have
u′= cos(x) and v′=−sin(x).
Step 3: Now, apply the product rule to find the derivative of f(x):
f′(x)=(u′v+uv′)
= (cos(x)·cos(x)) + (sin(x)·(−sin(x)))
= cos2(x)−sin2(x).
Hence, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos2(x)−sin2(x).
Question 2
Question
Find the derivative of the function f(x) = sin(x) cos(x).
Solution
Step 1: Apply the product rule, which states that if uand vare differentiable
functions of x, then the derivative of their product is (uv)′=u′v+uv′.
Step 2: Let u= sin(x) and v= cos(x).
Step 3: Find u′and v′.
Derivative of sin(x):
u′= cos(x)
Derivative of cos(x):
v′=−sin(x)
Step 4: Apply the product rule to find f′(x).
f′(x) = u′v+uv′
= (cos(x))(cos(x)) + (sin(x))(−sin(x))
= cos2(x)−sin2(x)
= cos(2x).
Therefore, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos(2x).
Question 3
Question
Find the derivative of the function f(x) = sin2(3x)−cos(2x) + tan(x).
Solution
Step 1: Apply the power rule and chain rule to find the derivative of sin2(3x).
f′(x) = d
dx (sin2(3x)) −
d
dx (cos(2x)) + d
dx (tan(x))
= 2 sin(3x) cos(3x)·3−(−sin(2x)·2) + sec2(x)
= 6 sin(3x) cos(3x) + 2 sin(2x) + sec2(x)
Therefore, the derivative of f(x) = sin2(3x)−cos(2x) + tan(x) is f′(x) =
6 sin(3x) cos(3x) + 2 sin(2x) + sec2(x).
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Question 4
Question
Find the derivative of the function f(x) = sin(2x) cos(3x).
Solution
To find the derivative of f(x), we will use the product rule along with the chain
rule for differentiation.
Step 1: Apply the product rule. Let u= sin(2x) and v= cos(3x).
f′(x) = u′v+uv′
Step 2: Find u′and v′.
u′= 2 cos(2x), v′=−3 sin(3x)
Step 3: Substitute into the product rule formula and simplify.
f′(x) = (2 cos(2x)·cos(3x)) + (sin(2x)· −3 sin(3x))
f′(x) = 2 cos(2x) cos(3x)−3 sin(2x) sin(3x)
Therefore, the derivative of f(x) = sin(2x) cos(3x) is f′(x) = 2 cos(2x) cos(3x)−
3 sin(2x) sin(3x).
Question 5
Question
Find the derivative of f(x) = sin2(x) cos(x).
Solution
To find the derivative of f(x) = sin2(x) cos(x), we will use the product rule and
the chain rule.
Step 1: Apply the product rule. Let u= sin2(x) and v= cos(x). Then,
f′(x) = u′v+uv′.
Step 2: Find u′and v′. Differentiating u= sin2(x) with respect to x, we
get
u′= 2 sin(x) cos(x).
Differentiating v= cos(x) with respect to x, we get
v′=−sin(x).
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Step 3: Substitute into the product rule formula. Substitute u,v,u′, and
v′into f′(x) = u′v+uv′to get
f′(x) = (2 sin(x) cos(x))(cos(x)) + (sin2(x))(−sin(x)).
Step 4: Simplify. Simplify the expression
f′(x) = 2 sin(x) cos2(x)−sin3(x).
Finally, the derivative of f(x) = sin2(x) cos(x) is f′(x) = 2 sin(x) cos2(x)−
sin3(x).
Question 6
Question
Find the derivative of f(x) = sin2(x) cos(x).
Solution
Step 1: Apply the product rule. Let u= sin2(x) and v= cos(x). Step 2: Find
u′and v′. Step 3: Apply the product rule to find f′(x). Step 4: Simplify the
expression for f′(x).
Step 1: Apply the product rule. Let u= sin2(x) and v= cos(x). Using the
product rule, we have
f′(x) = u′v+uv′.
Step 2: Find u′and v′. Taking the derivatives separately, we get
u′= 2 sin(x) cos(x)
and
v′=−sin(x).
Step 3: Apply the product rule to find f′(x). Substitute u,v,u′, and v′
into the product rule formula:
f′(x) = (2 sin(x) cos(x)) cos(x) + (sin2(x))(−sin(x)).
Step 4: Simplify the expression for f′(x). After simplifying, we get
f′(x) = 2 sin(x) cos2(x)−sin3(x).
Therefore, the derivative of f(x) = sin2(x) cos(x) is f′(x) = 2 sin(x) cos2(x)−
sin3(x).
Question 7
Question
Find the derivative of the function f(x) = sin(x)
1+cos(x).
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Solution
Step 1: To find the derivative of f(x), we will use the quotient rule which
states that if u(x) and v(x) are differentiable functions, then d
dx u(x)
v(x)=
v(x)u′(x)−u(x)v′(x)
(v(x))2.
Step 2: Identify u(x) and v(x) in the function f(x). Here, u(x) = sin(x) and
v(x) = 1 + cos(x).
Step 3: Find u′(x) and v′(x) by taking the derivatives of u(x) and v(x).
u′(x) = d
dx sin(x) = cos(x),
v′(x) = d
dx (1 + cos(x)) = −sin(x).
Step 4: Apply the quotient rule to find f′(x).
f′(x) = (1 + cos(x)) cos(x)−sin(x)(−sin(x))
(1 + cos(x))2
=cos(x) + cos2(x) + sin2(x)
(1 + cos(x))2
=cos(x)+1
(1 + cos(x))2.
Therefore, the derivative of f(x) = sin(x)
1+cos(x)is f′(x) = cos(x)+1
(1+cos(x))2.
Question 8
Question
Find the derivative of the function f(x) = sin2(3x) + cos(2x).
Solution
Step 1: Apply the chain rule to differentiate sin2(3x).
d
dx [sin2(3x)] = 2 sin(3x) cos(3x)·3 = 6 sin(3x) cos(3x)
Step 2: Differentiate cos(2x).
d
dx [cos(2x)] = −sin(2x)·2 = −2 sin(2x)
Step 3: Combine the derivatives from Steps 1 and 2 to find the derivative
of f(x).
f′(x) = 6 sin(3x) cos(3x)−2 sin(2x)
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Question 9
Question
Find the derivative of the function y=3 sin(x)−cos(x)
2 sin(x)+cos(x).
Solution
Step 1: To differentiate the given function, we will first rewrite it using the
quotient rule.
Step 2: The quotient rule states that for functions u(x) and v(x), the deriva-
tive of u(x)
v(x)is given by:
d
dx u(x)
v(x)=v(x)du
dx
−u(x)dv
dx
(v(x))2
Step 3: In this case, let u(x) = 3 sin(x)−cos(x) and v(x) = 2 sin(x)+cos(x).
Step 4: Calculate du
dx :
d
dx (3 sin(x)−cos(x)) = 3 cos(x) + sin(x)
Step 5: Calculate dv
dx :
d
dx (2 sin(x) + cos(x)) = 2 cos(x)−sin(x)
Step 6: Now, apply the quotient rule formula:
y′=(2 sin(x) + cos(x))(3 cos(x) + sin(x)) −(3 sin(x)−cos(x))(2 cos(x)−sin(x))
(2 sin(x) + cos(x))2
Step 7: Simplify the expression to find the derivative y′.
Step 8: After simplifying, the derivative of the function yis:
y′=
−5 sin(x)−2 cos(x)
(2 sin(x) + cos(x))2
Therefore, the derivative of the function y=3 sin(x)−cos(x)
2 sin(x)+cos(x)is y′=−5 sin(x)−2 cos(x)
(2 sin(x)+cos(x))2.
Question 10
Question
Find the derivative of the function f(x) = sin2(x)+cos2(x)
tan(x).
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Solution
Step 1: Simplify the function.
f(x) = sin2(x) + cos2(x)
tan(x)
=1
tan(x)
= cot(x).
Step 2: Differentiate f(x) = cot(x) with respect to xusing the quotient rule.
d
dx (cot(x)) = d
dx cos(x)
sin(x)
=(sin(x)(−sin(x)) −cos(x) cos(x))
sin2(x)
=
−sin2(x)−cos2(x)
sin2(x)
=
−(sin2(x) + cos2(x))
sin2(x)
=
−1
sin2(x)
=−csc2(x).
Question 11
Question
Find the derivative of the function f(x) = sin2(3x)−cos2(2x).
Solution
Step 1: Apply the chain rule to differentiate sin2(3x). Step 2: Apply the chain
rule to differentiate cos2(2x). Step 3: Simplify the derivative of each term. Step
4: Combine the derivatives to find the derivative of the given function.
Step 1: Let u= 3x. Then, the derivative of sin2(3x) using the chain rule
is: d
dx (sin2(3x)) = 2 sin(3x)·cos(3x)·3 = 6 sin(3x) cos(3x)
Step 2: Let v= 2x. Then, the derivative of cos2(2x) using the chain rule
is: d
dx (cos2(2x)) = −2 cos(2x) sin(2x)·2 = −4 cos(2x) sin(2x)
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Step 3: Simplify the derivatives we found in Step 1 and Step 2:
d
dx (sin2(3x)) = 6 sin(3x) cos(3x)
d
dx (cos2(2x)) = −4 cos(2x) sin(2x)
Step 4: Now, combine the derivatives to find the derivative of the given
function: d
dx (f(x)) = d
dx (sin2(3x)) −
d
dx (cos2(2x))
d
dx (f(x)) = 6 sin(3x) cos(3x)−(−4 cos(2x) sin(2x))
d
dx (f(x)) = 6 sin(3x) cos(3x) + 4 cos(2x) sin(2x)
Therefore, the derivative of the function f(x) = sin2(3x)−cos2(2x) is
6 sin(3x) cos(3x) + 4 cos(2x) sin(2x).
Question 12
Question
Find the derivative of the function f(x) = cos(x) sin(x).
Solution
Step 1: Apply the product rule to find the derivative of f(x). Step 2: Let
u(x) = cos(x) and v(x) = sin(x). Step 3: Find u′(x) and v′(x). Step 4: Apply
the product rule f′(x) = u(x)v′(x) + u′(x)v(x) to find f′(x). Step 5: Simplify
the result to obtain the final answer.
Question 13
Question
Find the derivative of f(x) = sin(2x) cos(3x).
Solution
Step 1: Use the product rule to differentiate f(x). Step 2: Apply the chain rule
to find the derivatives of sin(2x) and cos(3x). Step 3: Simplify the expression
by expanding and simplifying the trigonometric functions.
Step 1: Applying the product rule, we have:
f′(x) = (sin(2x))′(cos(3x)) + sin(2x)(cos(3x))′
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Step 2: Now, let’s find the derivatives of sin(2x) and cos(3x) using the chain
rule:
(sin(2x))′= 2 cos(2x)
(cos(3x))′=−3 sin(3x)
Step 3: Substitute these derivatives back into the expression for f′(x):
f′(x) = (2 cos(2x))(cos(3x)) + sin(2x)(−3 sin(3x))
f′(x) = 2 cos(2x) cos(3x)−3 sin(2x) sin(3x)
Therefore, the derivative of f(x) = sin(2x) cos(3x) is f′(x) = 2 cos(2x) cos(3x)−
3 sin(2x) sin(3x).
Question 14
Question
Find the derivative of the function f(x) = tan(x) sin(2x).
Solution
Step 1: Apply the product rule to find the derivative of f(x). Step 2: Recall the
product rule states that if f(x) = g(x)h(x), then f′(x) = g′(x)h(x) + g(x)h′(x).
Step 1: Let g(x) = tan(x) and h(x) = sin(2x). Then, using the product rule,
f′(x) = g′(x)h(x) + g(x)h′(x)
Step 2: Find the derivatives g′(x) and h′(x). - To find g′(x), differentiate
tan(x) with respect to x. Differentiating tan(x), we have:
g′(x) = sec2(x)
- To find h′(x), differentiate sin(2x) with respect to xusing the chain rule.
Let u= 2x, then du
dx = 2. Substitute u= 2xinto sin(u) to get:
h′(x) = 2 cos(2x)
Substitute g′(x), h(x), g(x), and h′(x) into the product rule formula:
f′(x) = sec2(x) sin(2x) + tan(x)·(2 cos(2x))
Therefore, the derivative of f(x) = tan(x) sin(2x) is
f′(x) = sec2(x) sin(2x) + 2 tan(x) cos(2x)
.
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Question 15
Question
Find the derivative of the function f(x) = cos2(3x) + sin2(4x).
Solution
Step 1: Recall the trigonometric identities cos2θ+ sin2θ= 1 and d
dx cos θ=
−sin θand d
dx sin θ= cos θ.
Step 2: Rewrite the function f(x) using the trigonometric identities:
f(x) = 1 + 1 = 2
Step 3: Find the derivative of f(x) using the constant multiple rule:
d
dx f(x) = d
dx 2=0
Step 4: Therefore, the derivative of the function f(x) = cos2(3x) + sin2(4x)
is 0 .
Question 16
Question
Find the derivative of the function f(x) = sin(x)−2 cos(x).
Solution
Step 1: Recall the derivatives of the trigonometric functions:
The derivative of sin(x) is cos(x).
The derivative of cos(x) is −sin(x).
Step 2: Find the derivative of f(x) = sin(x)−2 cos(x) using the sum/difference
rule for differentiation:
f′(x) = d
dx (sin(x)) −
d
dx (2 cos(x))
Step 3: Apply the derivatives of sin(x) and cos(x) that we recalled in Step
1:
f′(x) = cos(x)−2(−sin(x))
Step 4: Simplify the expression:
f′(x) = cos(x) + 2 sin(x)
Therefore, the derivative of the function f(x) = sin(x)−2 cos(x) is f′(x) =
cos(x) + 2 sin(x).
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Question 17
Question
Find the derivative of the function y= sin2(3x)−cos2(2x).
Solution
To find the derivative of the function y= sin2(3x)−cos2(2x), we will use the
chain rule and the derivative formulas for trigonometric functions.
Step 1: Find the derivative of sin2(3x).Let u= sin(3x). Then, the
derivative of sin2(3x) with respect to xcan be found using the chain rule:
d
dx (sin2(3x)) = 2 sin(3x) cos(3x)·3
= 6 sin(3x) cos(3x)
Step 2: Find the derivative of cos2(2x).Let v= cos(2x). Then, the
derivative of cos2(2x) with respect to xcan be found using the chain rule:
d
dx (cos2(2x)) = 2 cos(2x) sin(2x)·2
= 4 cos(2x) sin(2x)
Step 3: Combine the derivatives. Now, the derivative of y= sin2(3x)−
cos2(2x) is the sum of the derivatives we found in Step 1 and Step 2:
dy
dx = 6 sin(3x) cos(3x)−4 cos(2x) sin(2x)
Therefore, the derivative of y= sin2(3x)−cos2(2x) is 6 sin(3x) cos(3x)−
4 cos(2x) sin(2x).
Question 18
Question
Find the derivative of f(x) = sin(x)
cos(x).
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Solution
Step 1: Use the quotient rule to differentiate the function f(x) = sin(x)
cos(x).
f′(x) = d
dx sin(x)
cos(x)
=cos(x)·cos(x)−sin(x)·(−sin(x))
cos(x)2
=cos2(x) + sin2(x)
cos2(x)
=1
cos2(x)
= sec2(x)
So, the derivative of f(x) = sin(x)
cos(x)is f′(x) = sec2(x).
Question 19
Question
Find the derivative of y= sin2(3x) + cos(2x) with respect to x.
Solution
To find the derivative of y= sin2(3x) + cos(2x), we will use the chain rule
and derivative rules for trigonometric functions. Step 1: Find the derivative
of sin2(3x). Step 2: Find the derivative of cos(2x). Step 3: Combine the
derivatives to find dy
dx .
Step 1: Using the chain rule, let u= 3x:
d
dx (sin2(3x)) = d
du (sin2(u)) ·
du
dx
= 2 sin(3x) cos(3x)·3
= 6 sin(3x) cos(3x).
Step 2:
d
dx (cos(2x)) = −sin(2x)·2
=−2 sin(2x).
Step 3: Putting the derivatives together:
dy
dx = 6 sin(3x) cos(3x)−2 sin(2x)
= 3 sin(6x)−2 sin(2x).
So, the derivative of y= sin2(3x) + cos(2x) with respect to xis dy
dx =
3 sin(6x)−2 sin(2x).
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Question 20
Question
Find the derivative of y= sin(2x) cos(3x) with respect to x.
Solution
Step 1: Apply the product rule, which states that if uand vare functions of x,
then the derivative of their product is u′v+uv′.
Let u= sin(2x) and v= cos(3x). Then u′= 2 cos(2x) and v′=−3 sin(3x).
Step 2: Compute the derivative using the product rule as follows:
d
dx [sin(2x) cos(3x)] = u′v+uv′
= (2 cos(2x))(cos(3x)) + (sin(2x))(−3 sin(3x))
= 2 cos(2x) cos(3x)−3 sin(2x) sin(3x).
Therefore, the derivative of y= sin(2x) cos(3x) with respect to xis 2 cos(2x) cos(3x)−
3 sin(2x) sin(3x).
Question 21
Question
Find the derivative of the function f(x) = sin2(x)−cos2(x)
tan(x).
Solution
To find the derivative of the function f(x), we will first simplify the function
using trigonometric identities, then apply the quotient rule and chain rule to
find the derivative.
Step 1: Simplify the function using trigonometric identities.
f(x) = sin2(x)−cos2(x)
tan(x)
=sin2(x)−cos2(x)
sin(x)
cos(x)
= (sin2(x)−cos2(x)) ·
cos(x)
sin(x)
= sin(x) cos(x)−cos(x)
= sin(x)(cos(x)−1)
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Step 2: Use the product rule and chain rule to find the derivative of f(x).
Let u= sin(x) and v= cos(x)−1.
d
dx [sin(x)(cos(x)−1)] = du
dx
·v+u·
dv
dx
= (cos(x))(cos(x)−1) + sin(x)(−sin(x))
= cos2(x)−cos(x)−sin2(x)
Step 3: Simplify the result. Recall that cos2(x) + sin2(x) = 1. Therefore,
the derivative is:
cos2(x)−cos(x)−sin2(x) = 1 −cos(x)−1 = −cos(x)
Therefore, the derivative of the function f(x) is −cos(x).
Question 22
Question
Find the derivative of the function f(x) = sin2(x) cos(x).
Solution
Step 1: Use the product rule to differentiate f(x) = sin2(x) cos(x). Step 2: Let
u= sin2(x) and v= cos(x). Step 3: Find u′and v′.
u′=d
dx (sin2(x))
= 2 sin(x) cos(x)
= 2 sin(x) cos(x)
v′=d
dx (cos(x))
=−sin(x)
Step 4: Apply the product rule f′(x) = u′v+uv′.
f′(x) = (2 sin(x) cos(x)) cos(x) + sin2(x)(−sin(x))
= 2 sin(x) cos2(x)−sin3(x)
Therefore, the derivative of f(x) = sin2(x) cos(x) is f′(x) = 2 sin(x) cos2(x)−
sin3(x).
Question 23
Question
Find the derivative of f(x) = sin(x) cos(x).
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Solution
To find the derivative of f(x) = sin(x) cos(x), we will use the product rule,
which states that if h(x) = g(x)·j(x), then h′(x) = g′(x)·j(x) + g(x)·j′(x).
Step 1: Let g(x) = sin(x) and j(x) = cos(x). Then, we have g′(x) = cos(x)
and j′(x) = −sin(x).
Step 2: Apply the product rule to find f′(x):
f′(x) = g′(x)·j(x) + g(x)·j′(x)
= cos(x)·cos(x) + sin(x)·(−sin(x))
= cos2(x)−sin2(x)
= cos(2x).
Therefore, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos(2x).
Question 24
Question
Find the derivative of the function f(x) = sin(x) cos(x).
Solution
Step 1: Apply the product rule, d
dx (u·v) = u′
·v+u·v′, where u= sin(x) and
v= cos(x). Step 2: Calculate the derivatives of uand v. Step 3: Let’s find u′
and v′. Step 4:
u′=d
dx (sin(x)) = cos(x)
v′=d
dx (cos(x)) = −sin(x)
Step 5: Substitute u′,v′,u, and vinto the product rule formula. Step 6: So,
d
dx (sin(x) cos(x)) = cos(x) cos(x) + sin(x)·(−sin(x))
= cos2(x)−sin2(x)
Therefore, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos2(x)−sin2(x).
Question 25
Question
Differentiate the function f(x) = sin2(2x) + sin2(x) with respect to x.
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Solution
Step 1: Apply the chain rule to differentiate sin2(2x). Step 2: Apply the chain
rule to differentiate sin2(x). Step 3: Combine the results to find the derivative
of f(x).
Step 1: Let u= sin(2x), then f(x) = u2. Using the chain rule, we have
df
dx =d
dx (sin(2x))2= 2 sin(2x) cos(2x)·2 = 4 sin(2x) cos(2x).
Step 2: Let v= sin(x), then g(x) = v2. Applying the chain rule, we get
dg
dx =d
dx (sin(x))2= 2 sin(x) cos(x) = 2 sin(x) cos(x).
Step 3: Now, differentiate the function f(x) = sin2(2x) + sin2(x):
f′(x) = 4 sin(2x) cos(2x) + 2 sin(x) cos(x).
Therefore, the derivative of f(x) with respect to xis f′(x) = 4 sin(2x) cos(2x)+
2 sin(x) cos(x).
Question 26
Question
Compute the derivative of f(x) = tan(x)·cos(x).
Solution
Step 1: We will use the product rule to differentiate f(x), which states that if
f(x) = u(x)·v(x), then f′(x) = u(x)·v′(x) + u′(x)·v(x).
Step 2: Let u(x) = tan(x) and v(x) = cos(x).
Step 3: Find u′(x) and v′(x).
u′(x) = d
dx (tan(x)) = sec2(x) by using the derivative of tan(x).
v′(x) = d
dx (cos(x)) = −sin(x) by using the derivative of cos(x).
Step 4: Apply the product rule to find f′(x).
f′(x) = tan(x)·(−sin(x)) + sec2(x)·cos(x)
Step 5: Simplify the expression.
f′(x) = −tan(x)·sin(x) + sec2(x)·cos(x)
Step 6: Thus, the derivative of f(x) = tan(x)·cos(x) is f′(x) = −tan(x)·
sin(x) + sec2(x)·cos(x).
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Question 27
Question
Find the derivative of the function f(x) = sin(2x)+3 cos(x)
x.
Solution
Step 1: To find the derivative of f(x), we will use the quotient rule. The quotient
rule states that if f(x) = g(x)
h(x), then f′(x) = g′(x)h(x)−g(x)h′(x)
(h(x))2.
Step 2: Let g(x) = sin(2x) + 3 cos(x) and h(x) = x. We can find the
derivatives of g(x) and h(x):
For g(x):
g′(x) = d
dx (sin(2x) + 3 cos(x))
g′(x) = 2 cos(2x)−3 sin(x)
For h(x):
h′(x)=1
Step 3: Now, we can apply the quotient rule to find f′(x):
f′(x) = (2 cos(2x)−3 sin(x))(x)−(sin(2x) + 3 cos(x))(1)
x2
f′(x) = 2xcos(2x)−3xsin(x)−sin(2x)−3 cos(x)
x2
Therefore, the derivative of the function f(x) = sin(2x)+3 cos(x)
xis f′(x) =
2xcos(2x)−3xsin(x)−sin(2x)−3 cos(x)
x2.
Question 28
Question
Compute the derivative of f(x) = 3 sin(2x) + 4 cos(3x).
Solution
Step 1: Apply the derivative formulas for sine and cosine functions.
Step 2: Compute the derivative of f(x) using the chain rule.
Step 3: Simplify the final result.
Step 1:
The derivative of sin(ax) is acos(ax) and the derivative of cos(ax) is −asin(ax).
Therefore, the derivative of f(x) is:
f′(x) = 3(2 cos(2x)) + 4(−3 sin(3x))
17
Step 2:
Applying the chain rule, we have:
f′(x) = 6 cos(2x)−12 sin(3x)
Step 3:
Simplify the final result:
f′(x) = 6 cos(2x)−12 sin(3x)
Question 29
Question
Find the derivative of the function f(x) = cos2(2x) + sin2(3x).
Solution
Step 1: Apply the chain rule to differentiate cos2(2x).
d
dx (cos2(2x)) = 2 cos(2x)·
d
dx (cos(2x))
Step 2: Apply the chain rule and power rule to find d
dx (cos(2x)).
d
dx (cos(2x)) = −2 sin(2x)
Step 3: Put the results from Step 1 and Step 2 together to differentiate
cos2(2x).
d
dx (cos2(2x)) = 2 cos(2x)·(−2 sin(2x)) = −4 cos(2x) sin(2x)
Step 4: Apply the same steps to differentiate sin2(3x).
d
dx (sin2(3x)) = 2 sin(3x)·3 cos(3x) = 6 sin(3x) cos(3x)
Step 5: Put the results from Step 3 and Step 4 together to find the derivative
of f(x).
f′(x) = −4 cos(2x) sin(2x) + 6 sin(3x) cos(3x)
Question 30
Question
Find the derivative of y= sin2(3x) + cos(2x) with respect to x.
18
Solution
Step 1: Use the chain rule to differentiate sin2(3x).
d
dx (sin2(3x)) = 2 sin(3x) cos(3x)·3 = 6 sin(3x) cos(3x)
Step 2: Differentiate cos(2x).
d
dx (cos(2x)) = −sin(2x)·2 = −2 sin(2x)
Step 3: The derivative of y= sin2(3x) + cos(2x) with respect to xis:
d
dx (y) = 6 sin(3x) cos(3x)−2 sin(2x)
Question 31
Question
Find the derivative of the function f(x) = sin2(3x) + cos3(2x).
Solution
Step 1: Apply the chain rule and power rule to differentiate sin2(3x).
f′(x) = 2 sin(3x)·cos(3x)·3
= 6 sin(3x) cos(3x).
Step 2: Apply the chain rule and power rule to differentiate cos3(2x).
f′(x) = −3 cos2(2x)·(−sin(2x)) ·2
= 6 cos2(2x) sin(2x).
Step 3: Add the derivatives of the two terms to find f′(x).
f′(x) = 6 sin(3x) cos(3x) + 6 cos2(2x) sin(2x).
Question 32
Question
Find the derivative of the function f(x) = sin(x) cos(x).
19
Solution
Step 1: Apply the product rule, (f g)′=f′g+fg′, where f(x) = sin(x) and
g(x) = cos(x). Step 2: Find f′(x) and g′(x). Step 3: Calculate the derivative of
f(x) = sin(x). Step 4: Use the derivative formula for sin(x), d
dx sin(x) = cos(x).
Step 5: Therefore, f′(x) = cos(x). Step 6: Calculate the derivative of g(x) =
cos(x). Step 7: Use the derivative formula for cos(x), d
dx cos(x) = −sin(x). Step
8: Thus, g′(x) = −sin(x). Step 9: Apply the product rule to find f(x)g′(x) +
f′(x)g(x). Step 10: Substitute f′(x) = cos(x) and g′(x) = −sin(x) into the
product rule. Step 11: Simplify the expression to get the final answer. Step 12:
The derivative of the function f(x) = sin(x) cos(x) is f′(x) = cos2(x)−sin2(x).
Question 33
Question
Find the derivative of y= sin2(5x3−2x) with respect to x.
Solution
Step 1: Apply chain rule - differentiate the outer function first, then the inner
function.
dy
dx = 2 sin5x3−2xcos5x3−2x(15x2−2)
= 2 sin5x3−2xcos5x3−2x(15x2−2)
Step 2: Simplify the expression by combining terms.
dy
dx = 30x2sin5x3−2xcos5x3−2x−4 sin5x3−2xcos5x3−2x
Step 3: Factor out sin5x3−2xcos5x3−2x.
dy
dx = (30x2−4) sin5x3−2xcos5x3−2x
Therefore, the derivative of y= sin2(5x3−2x) with respect to xis (30x2−
4) sin5x3−2xcos5x3−2x.
Question 34
Question
Find the derivative of the function f(x) = sin(x) cos(x).
20
Solution
To find the derivative of the function f(x) = sin(x) cos(x), we will use the
product rule of differentiation.
Step 1: Apply the product rule. Let u= sin(x) and v= cos(x).
f′(x) = u′v+uv′
= (sin′(x) cos(x) + sin(x) cos′(x))
= (cos(x) cos(x) + sin(x)(−sin(x)))
= cos2(x)−sin2(x)
= cos(2x).
Step 2: Simplify the final expression. The derivative of f(x) = sin(x) cos(x)
is f′(x) = cos(2x).
Question 35
Question
Find the derivative of the function f(x) = cos(3x) sin(4x).
Solution
Step 1: Apply the product rule, u′v+uv′, where u= cos(3x) and v= sin(4x).
Step 2: Find u′and v′.
u′=−3 sin(3x),
v′= 4 cos(4x).
Step 3: Apply the product rule formula.
f′(x) = u′v+uv′
= (−3 sin(3x))(sin(4x)) + (cos(3x))(4 cos(4x))
=−3 sin(3x) sin(4x) + 4 cos(3x) cos(4x).
Therefore, the derivative of the function f(x) = cos(3x) sin(4x) is f′(x) =
−3 sin(3x) sin(4x) + 4 cos(3x) cos(4x).
21
Question 2
Question
Find the derivative of the function f(x) = sin(x) cos(x).
Solution
Step 1: Apply the product rule, which states that if uand vare differentiable
functions of x, then the derivative of their product is (uv)′=u′v+uv′.
Step 2: Let u= sin(x) and v= cos(x).
Step 3: Find u′and v′.
Derivative of sin(x):
u′= cos(x)
Derivative of cos(x):
v′=−sin(x)
Step 4: Apply the product rule to find f′(x).
f′(x) = u′v+uv′
= (cos(x))(cos(x)) + (sin(x))(−sin(x))
= cos2(x)−sin2(x)
= cos(2x).
Therefore, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos(2x).
Question 3
Question
Find the derivative of the function f(x) = sin2(3x)−cos(2x) + tan(x).
Solution
Step 1: Apply the power rule and chain rule to find the derivative of sin2(3x).
f′(x) = d
dx (sin2(3x)) −
d
dx (cos(2x)) + d
dx (tan(x))
= 2 sin(3x) cos(3x)·3−(−sin(2x)·2) + sec2(x)
= 6 sin(3x) cos(3x) + 2 sin(2x) + sec2(x)
Therefore, the derivative of f(x) = sin2(3x)−cos(2x) + tan(x) is f′(x) =
6 sin(3x) cos(3x) + 2 sin(2x) + sec2(x).
2
Question 4
Question
Find the derivative of the function f(x) = sin(2x) cos(3x).
Solution
To find the derivative of f(x), we will use the product rule along with the chain
rule for differentiation.
Step 1: Apply the product rule. Let u= sin(2x) and v= cos(3x).
f′(x) = u′v+uv′
Step 2: Find u′and v′.
u′= 2 cos(2x), v′=−3 sin(3x)
Step 3: Substitute into the product rule formula and simplify.
f′(x) = (2 cos(2x)·cos(3x)) + (sin(2x)· −3 sin(3x))
f′(x) = 2 cos(2x) cos(3x)−3 sin(2x) sin(3x)
Therefore, the derivative of f(x) = sin(2x) cos(3x) is f′(x) = 2 cos(2x) cos(3x)−
3 sin(2x) sin(3x).
Question 5
Question
Find the derivative of f(x) = sin2(x) cos(x).
Solution
To find the derivative of f(x) = sin2(x) cos(x), we will use the product rule and
the chain rule.
Step 1: Apply the product rule. Let u= sin2(x) and v= cos(x). Then,
f′(x) = u′v+uv′.
Step 2: Find u′and v′. Differentiating u= sin2(x) with respect to x, we
get
u′= 2 sin(x) cos(x).
Differentiating v= cos(x) with respect to x, we get
v′=−sin(x).
3
Step 3: Substitute into the product rule formula. Substitute u,v,u′, and
v′into f′(x) = u′v+uv′to get
f′(x) = (2 sin(x) cos(x))(cos(x)) + (sin2(x))(−sin(x)).
Step 4: Simplify. Simplify the expression
f′(x) = 2 sin(x) cos2(x)−sin3(x).
Finally, the derivative of f(x) = sin2(x) cos(x) is f′(x) = 2 sin(x) cos2(x)−
sin3(x).
Question 6
Question
Find the derivative of f(x) = sin2(x) cos(x).
Solution
Step 1: Apply the product rule. Let u= sin2(x) and v= cos(x). Step 2: Find
u′and v′. Step 3: Apply the product rule to find f′(x). Step 4: Simplify the
expression for f′(x).
Step 1: Apply the product rule. Let u= sin2(x) and v= cos(x). Using the
product rule, we have
f′(x) = u′v+uv′.
Step 2: Find u′and v′. Taking the derivatives separately, we get
u′= 2 sin(x) cos(x)
and
v′=−sin(x).
Step 3: Apply the product rule to find f′(x). Substitute u,v,u′, and v′
into the product rule formula:
f′(x) = (2 sin(x) cos(x)) cos(x) + (sin2(x))(−sin(x)).
Step 4: Simplify the expression for f′(x). After simplifying, we get
f′(x) = 2 sin(x) cos2(x)−sin3(x).
Therefore, the derivative of f(x) = sin2(x) cos(x) is f′(x) = 2 sin(x) cos2(x)−
sin3(x).
Question 7
Question
Find the derivative of the function f(x) = sin(x)
1+cos(x).
4
Solution
Step 1: To find the derivative of f(x), we will use the quotient rule which
states that if u(x) and v(x) are differentiable functions, then d
dx u(x)
v(x)=
v(x)u′(x)−u(x)v′(x)
(v(x))2.
Step 2: Identify u(x) and v(x) in the function f(x). Here, u(x) = sin(x) and
v(x) = 1 + cos(x).
Step 3: Find u′(x) and v′(x) by taking the derivatives of u(x) and v(x).
u′(x) = d
dx sin(x) = cos(x),
v′(x) = d
dx (1 + cos(x)) = −sin(x).
Step 4: Apply the quotient rule to find f′(x).
f′(x) = (1 + cos(x)) cos(x)−sin(x)(−sin(x))
(1 + cos(x))2
=cos(x) + cos2(x) + sin2(x)
(1 + cos(x))2
=cos(x)+1
(1 + cos(x))2.
Therefore, the derivative of f(x) = sin(x)
1+cos(x)is f′(x) = cos(x)+1
(1+cos(x))2.
Question 8
Question
Find the derivative of the function f(x) = sin2(3x) + cos(2x).
Solution
Step 1: Apply the chain rule to differentiate sin2(3x).
d
dx [sin2(3x)] = 2 sin(3x) cos(3x)·3 = 6 sin(3x) cos(3x)
Step 2: Differentiate cos(2x).
d
dx [cos(2x)] = −sin(2x)·2 = −2 sin(2x)
Step 3: Combine the derivatives from Steps 1 and 2 to find the derivative
of f(x).
f′(x) = 6 sin(3x) cos(3x)−2 sin(2x)
5
Question 9
Question
Find the derivative of the function y=3 sin(x)−cos(x)
2 sin(x)+cos(x).
Solution
Step 1: To differentiate the given function, we will first rewrite it using the
quotient rule.
Step 2: The quotient rule states that for functions u(x) and v(x), the deriva-
tive of u(x)
v(x)is given by:
d
dx u(x)
v(x)=v(x)du
dx
−u(x)dv
dx
(v(x))2
Step 3: In this case, let u(x) = 3 sin(x)−cos(x) and v(x) = 2 sin(x)+cos(x).
Step 4: Calculate du
dx :
d
dx (3 sin(x)−cos(x)) = 3 cos(x) + sin(x)
Step 5: Calculate dv
dx :
d
dx (2 sin(x) + cos(x)) = 2 cos(x)−sin(x)
Step 6: Now, apply the quotient rule formula:
y′=(2 sin(x) + cos(x))(3 cos(x) + sin(x)) −(3 sin(x)−cos(x))(2 cos(x)−sin(x))
(2 sin(x) + cos(x))2
Step 7: Simplify the expression to find the derivative y′.
Step 8: After simplifying, the derivative of the function yis:
y′=
−5 sin(x)−2 cos(x)
(2 sin(x) + cos(x))2
Therefore, the derivative of the function y=3 sin(x)−cos(x)
2 sin(x)+cos(x)is y′=−5 sin(x)−2 cos(x)
(2 sin(x)+cos(x))2.
Question 10
Question
Find the derivative of the function f(x) = sin2(x)+cos2(x)
tan(x).
6
Solution
Step 1: Simplify the function.
f(x) = sin2(x) + cos2(x)
tan(x)
=1
tan(x)
= cot(x).
Step 2: Differentiate f(x) = cot(x) with respect to xusing the quotient rule.
d
dx (cot(x)) = d
dx cos(x)
sin(x)
=(sin(x)(−sin(x)) −cos(x) cos(x))
sin2(x)
=
−sin2(x)−cos2(x)
sin2(x)
=
−(sin2(x) + cos2(x))
sin2(x)
=
−1
sin2(x)
=−csc2(x).
Question 11
Question
Find the derivative of the function f(x) = sin2(3x)−cos2(2x).
Solution
Step 1: Apply the chain rule to differentiate sin2(3x). Step 2: Apply the chain
rule to differentiate cos2(2x). Step 3: Simplify the derivative of each term. Step
4: Combine the derivatives to find the derivative of the given function.
Step 1: Let u= 3x. Then, the derivative of sin2(3x) using the chain rule
is: d
dx (sin2(3x)) = 2 sin(3x)·cos(3x)·3 = 6 sin(3x) cos(3x)
Step 2: Let v= 2x. Then, the derivative of cos2(2x) using the chain rule
is: d
dx (cos2(2x)) = −2 cos(2x) sin(2x)·2 = −4 cos(2x) sin(2x)
7
Step 3: Simplify the derivatives we found in Step 1 and Step 2:
d
dx (sin2(3x)) = 6 sin(3x) cos(3x)
d
dx (cos2(2x)) = −4 cos(2x) sin(2x)
Step 4: Now, combine the derivatives to find the derivative of the given
function: d
dx (f(x)) = d
dx (sin2(3x)) −
d
dx (cos2(2x))
d
dx (f(x)) = 6 sin(3x) cos(3x)−(−4 cos(2x) sin(2x))
d
dx (f(x)) = 6 sin(3x) cos(3x) + 4 cos(2x) sin(2x)
Therefore, the derivative of the function f(x) = sin2(3x)−cos2(2x) is
6 sin(3x) cos(3x) + 4 cos(2x) sin(2x).
Question 12
Question
Find the derivative of the function f(x) = cos(x) sin(x).
Solution
Step 1: Apply the product rule to find the derivative of f(x). Step 2: Let
u(x) = cos(x) and v(x) = sin(x). Step 3: Find u′(x) and v′(x). Step 4: Apply
the product rule f′(x) = u(x)v′(x) + u′(x)v(x) to find f′(x). Step 5: Simplify
the result to obtain the final answer.
Question 13
Question
Find the derivative of f(x) = sin(2x) cos(3x).
Solution
Step 1: Use the product rule to differentiate f(x). Step 2: Apply the chain rule
to find the derivatives of sin(2x) and cos(3x). Step 3: Simplify the expression
by expanding and simplifying the trigonometric functions.
Step 1: Applying the product rule, we have:
f′(x) = (sin(2x))′(cos(3x)) + sin(2x)(cos(3x))′
8
Step 2: Now, let’s find the derivatives of sin(2x) and cos(3x) using the chain
rule:
(sin(2x))′= 2 cos(2x)
(cos(3x))′=−3 sin(3x)
Step 3: Substitute these derivatives back into the expression for f′(x):
f′(x) = (2 cos(2x))(cos(3x)) + sin(2x)(−3 sin(3x))
f′(x) = 2 cos(2x) cos(3x)−3 sin(2x) sin(3x)
Therefore, the derivative of f(x) = sin(2x) cos(3x) is f′(x) = 2 cos(2x) cos(3x)−
3 sin(2x) sin(3x).
Question 14
Question
Find the derivative of the function f(x) = tan(x) sin(2x).
Solution
Step 1: Apply the product rule to find the derivative of f(x). Step 2: Recall the
product rule states that if f(x) = g(x)h(x), then f′(x) = g′(x)h(x) + g(x)h′(x).
Step 1: Let g(x) = tan(x) and h(x) = sin(2x). Then, using the product rule,
f′(x) = g′(x)h(x) + g(x)h′(x)
Step 2: Find the derivatives g′(x) and h′(x). - To find g′(x), differentiate
tan(x) with respect to x. Differentiating tan(x), we have:
g′(x) = sec2(x)
- To find h′(x), differentiate sin(2x) with respect to xusing the chain rule.
Let u= 2x, then du
dx = 2. Substitute u= 2xinto sin(u) to get:
h′(x) = 2 cos(2x)
Substitute g′(x), h(x), g(x), and h′(x) into the product rule formula:
f′(x) = sec2(x) sin(2x) + tan(x)·(2 cos(2x))
Therefore, the derivative of f(x) = tan(x) sin(2x) is
f′(x) = sec2(x) sin(2x) + 2 tan(x) cos(2x)
.
9
Question 15
Question
Find the derivative of the function f(x) = cos2(3x) + sin2(4x).
Solution
Step 1: Recall the trigonometric identities cos2θ+ sin2θ= 1 and d
dx cos θ=
−sin θand d
dx sin θ= cos θ.
Step 2: Rewrite the function f(x) using the trigonometric identities:
f(x) = 1 + 1 = 2
Step 3: Find the derivative of f(x) using the constant multiple rule:
d
dx f(x) = d
dx 2=0
Step 4: Therefore, the derivative of the function f(x) = cos2(3x) + sin2(4x)
is 0 .
Question 16
Question
Find the derivative of the function f(x) = sin(x)−2 cos(x).
Solution
Step 1: Recall the derivatives of the trigonometric functions:
The derivative of sin(x) is cos(x).
The derivative of cos(x) is −sin(x).
Step 2: Find the derivative of f(x) = sin(x)−2 cos(x) using the sum/difference
rule for differentiation:
f′(x) = d
dx (sin(x)) −
d
dx (2 cos(x))
Step 3: Apply the derivatives of sin(x) and cos(x) that we recalled in Step
1:
f′(x) = cos(x)−2(−sin(x))
Step 4: Simplify the expression:
f′(x) = cos(x) + 2 sin(x)
Therefore, the derivative of the function f(x) = sin(x)−2 cos(x) is f′(x) =
cos(x) + 2 sin(x).
10
Question 17
Question
Find the derivative of the function y= sin2(3x)−cos2(2x).
Solution
To find the derivative of the function y= sin2(3x)−cos2(2x), we will use the
chain rule and the derivative formulas for trigonometric functions.
Step 1: Find the derivative of sin2(3x).Let u= sin(3x). Then, the
derivative of sin2(3x) with respect to xcan be found using the chain rule:
d
dx (sin2(3x)) = 2 sin(3x) cos(3x)·3
= 6 sin(3x) cos(3x)
Step 2: Find the derivative of cos2(2x).Let v= cos(2x). Then, the
derivative of cos2(2x) with respect to xcan be found using the chain rule:
d
dx (cos2(2x)) = 2 cos(2x) sin(2x)·2
= 4 cos(2x) sin(2x)
Step 3: Combine the derivatives. Now, the derivative of y= sin2(3x)−
cos2(2x) is the sum of the derivatives we found in Step 1 and Step 2:
dy
dx = 6 sin(3x) cos(3x)−4 cos(2x) sin(2x)
Therefore, the derivative of y= sin2(3x)−cos2(2x) is 6 sin(3x) cos(3x)−
4 cos(2x) sin(2x).
Question 18
Question
Find the derivative of f(x) = sin(x)
cos(x).
11
Solution
Step 1: Use the quotient rule to differentiate the function f(x) = sin(x)
cos(x).
f′(x) = d
dx sin(x)
cos(x)
=cos(x)·cos(x)−sin(x)·(−sin(x))
cos(x)2
=cos2(x) + sin2(x)
cos2(x)
=1
cos2(x)
= sec2(x)
So, the derivative of f(x) = sin(x)
cos(x)is f′(x) = sec2(x).
Question 19
Question
Find the derivative of y= sin2(3x) + cos(2x) with respect to x.
Solution
To find the derivative of y= sin2(3x) + cos(2x), we will use the chain rule
and derivative rules for trigonometric functions. Step 1: Find the derivative
of sin2(3x). Step 2: Find the derivative of cos(2x). Step 3: Combine the
derivatives to find dy
dx .
Step 1: Using the chain rule, let u= 3x:
d
dx (sin2(3x)) = d
du (sin2(u)) ·
du
dx
= 2 sin(3x) cos(3x)·3
= 6 sin(3x) cos(3x).
Step 2:
d
dx (cos(2x)) = −sin(2x)·2
=−2 sin(2x).
Step 3: Putting the derivatives together:
dy
dx = 6 sin(3x) cos(3x)−2 sin(2x)
= 3 sin(6x)−2 sin(2x).
So, the derivative of y= sin2(3x) + cos(2x) with respect to xis dy
dx =
3 sin(6x)−2 sin(2x).
12
Question 20
Question
Find the derivative of y= sin(2x) cos(3x) with respect to x.
Solution
Step 1: Apply the product rule, which states that if uand vare functions of x,
then the derivative of their product is u′v+uv′.
Let u= sin(2x) and v= cos(3x). Then u′= 2 cos(2x) and v′=−3 sin(3x).
Step 2: Compute the derivative using the product rule as follows:
d
dx [sin(2x) cos(3x)] = u′v+uv′
= (2 cos(2x))(cos(3x)) + (sin(2x))(−3 sin(3x))
= 2 cos(2x) cos(3x)−3 sin(2x) sin(3x).
Therefore, the derivative of y= sin(2x) cos(3x) with respect to xis 2 cos(2x) cos(3x)−
3 sin(2x) sin(3x).
Question 21
Question
Find the derivative of the function f(x) = sin2(x)−cos2(x)
tan(x).
Solution
To find the derivative of the function f(x), we will first simplify the function
using trigonometric identities, then apply the quotient rule and chain rule to
find the derivative.
Step 1: Simplify the function using trigonometric identities.
f(x) = sin2(x)−cos2(x)
tan(x)
=sin2(x)−cos2(x)
sin(x)
cos(x)
= (sin2(x)−cos2(x)) ·
cos(x)
sin(x)
= sin(x) cos(x)−cos(x)
= sin(x)(cos(x)−1)
13
Step 2: Use the product rule and chain rule to find the derivative of f(x).
Let u= sin(x) and v= cos(x)−1.
d
dx [sin(x)(cos(x)−1)] = du
dx
·v+u·
dv
dx
= (cos(x))(cos(x)−1) + sin(x)(−sin(x))
= cos2(x)−cos(x)−sin2(x)
Step 3: Simplify the result. Recall that cos2(x) + sin2(x) = 1. Therefore,
the derivative is:
cos2(x)−cos(x)−sin2(x) = 1 −cos(x)−1 = −cos(x)
Therefore, the derivative of the function f(x) is −cos(x).
Question 22
Question
Find the derivative of the function f(x) = sin2(x) cos(x).
Solution
Step 1: Use the product rule to differentiate f(x) = sin2(x) cos(x). Step 2: Let
u= sin2(x) and v= cos(x). Step 3: Find u′and v′.
u′=d
dx (sin2(x))
= 2 sin(x) cos(x)
= 2 sin(x) cos(x)
v′=d
dx (cos(x))
=−sin(x)
Step 4: Apply the product rule f′(x) = u′v+uv′.
f′(x) = (2 sin(x) cos(x)) cos(x) + sin2(x)(−sin(x))
= 2 sin(x) cos2(x)−sin3(x)
Therefore, the derivative of f(x) = sin2(x) cos(x) is f′(x) = 2 sin(x) cos2(x)−
sin3(x).
Question 23
Question
Find the derivative of f(x) = sin(x) cos(x).
14
Solution
To find the derivative of f(x) = sin(x) cos(x), we will use the product rule,
which states that if h(x) = g(x)·j(x), then h′(x) = g′(x)·j(x) + g(x)·j′(x).
Step 1: Let g(x) = sin(x) and j(x) = cos(x). Then, we have g′(x) = cos(x)
and j′(x) = −sin(x).
Step 2: Apply the product rule to find f′(x):
f′(x) = g′(x)·j(x) + g(x)·j′(x)
= cos(x)·cos(x) + sin(x)·(−sin(x))
= cos2(x)−sin2(x)
= cos(2x).
Therefore, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos(2x).
Question 24
Question
Find the derivative of the function f(x) = sin(x) cos(x).
Solution
Step 1: Apply the product rule, d
dx (u·v) = u′
·v+u·v′, where u= sin(x) and
v= cos(x). Step 2: Calculate the derivatives of uand v. Step 3: Let’s find u′
and v′. Step 4:
u′=d
dx (sin(x)) = cos(x)
v′=d
dx (cos(x)) = −sin(x)
Step 5: Substitute u′,v′,u, and vinto the product rule formula. Step 6: So,
d
dx (sin(x) cos(x)) = cos(x) cos(x) + sin(x)·(−sin(x))
= cos2(x)−sin2(x)
Therefore, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos2(x)−sin2(x).
Question 25
Question
Differentiate the function f(x) = sin2(2x) + sin2(x) with respect to x.
15
Solution
Step 1: Apply the chain rule to differentiate sin2(2x). Step 2: Apply the chain
rule to differentiate sin2(x). Step 3: Combine the results to find the derivative
of f(x).
Step 1: Let u= sin(2x), then f(x) = u2. Using the chain rule, we have
df
dx =d
dx (sin(2x))2= 2 sin(2x) cos(2x)·2 = 4 sin(2x) cos(2x).
Step 2: Let v= sin(x), then g(x) = v2. Applying the chain rule, we get
dg
dx =d
dx (sin(x))2= 2 sin(x) cos(x) = 2 sin(x) cos(x).
Step 3: Now, differentiate the function f(x) = sin2(2x) + sin2(x):
f′(x) = 4 sin(2x) cos(2x) + 2 sin(x) cos(x).
Therefore, the derivative of f(x) with respect to xis f′(x) = 4 sin(2x) cos(2x)+
2 sin(x) cos(x).
Question 26
Question
Compute the derivative of f(x) = tan(x)·cos(x).
Solution
Step 1: We will use the product rule to differentiate f(x), which states that if
f(x) = u(x)·v(x), then f′(x) = u(x)·v′(x) + u′(x)·v(x).
Step 2: Let u(x) = tan(x) and v(x) = cos(x).
Step 3: Find u′(x) and v′(x).
u′(x) = d
dx (tan(x)) = sec2(x) by using the derivative of tan(x).
v′(x) = d
dx (cos(x)) = −sin(x) by using the derivative of cos(x).
Step 4: Apply the product rule to find f′(x).
f′(x) = tan(x)·(−sin(x)) + sec2(x)·cos(x)
Step 5: Simplify the expression.
f′(x) = −tan(x)·sin(x) + sec2(x)·cos(x)
Step 6: Thus, the derivative of f(x) = tan(x)·cos(x) is f′(x) = −tan(x)·
sin(x) + sec2(x)·cos(x).
16
Question 27
Question
Find the derivative of the function f(x) = sin(2x)+3 cos(x)
x.
Solution
Step 1: To find the derivative of f(x), we will use the quotient rule. The quotient
rule states that if f(x) = g(x)
h(x), then f′(x) = g′(x)h(x)−g(x)h′(x)
(h(x))2.
Step 2: Let g(x) = sin(2x) + 3 cos(x) and h(x) = x. We can find the
derivatives of g(x) and h(x):
For g(x):
g′(x) = d
dx (sin(2x) + 3 cos(x))
g′(x) = 2 cos(2x)−3 sin(x)
For h(x):
h′(x)=1
Step 3: Now, we can apply the quotient rule to find f′(x):
f′(x) = (2 cos(2x)−3 sin(x))(x)−(sin(2x) + 3 cos(x))(1)
x2
f′(x) = 2xcos(2x)−3xsin(x)−sin(2x)−3 cos(x)
x2
Therefore, the derivative of the function f(x) = sin(2x)+3 cos(x)
xis f′(x) =
2xcos(2x)−3xsin(x)−sin(2x)−3 cos(x)
x2.
Question 28
Question
Compute the derivative of f(x) = 3 sin(2x) + 4 cos(3x).
Solution
Step 1: Apply the derivative formulas for sine and cosine functions.
Step 2: Compute the derivative of f(x) using the chain rule.
Step 3: Simplify the final result.
Step 1:
The derivative of sin(ax) is acos(ax) and the derivative of cos(ax) is −asin(ax).
Therefore, the derivative of f(x) is:
f′(x) = 3(2 cos(2x)) + 4(−3 sin(3x))
17
Step 2:
Applying the chain rule, we have:
f′(x) = 6 cos(2x)−12 sin(3x)
Step 3:
Simplify the final result:
f′(x) = 6 cos(2x)−12 sin(3x)
Question 29
Question
Find the derivative of the function f(x) = cos2(2x) + sin2(3x).
Solution
Step 1: Apply the chain rule to differentiate cos2(2x).
d
dx (cos2(2x)) = 2 cos(2x)·
d
dx (cos(2x))
Step 2: Apply the chain rule and power rule to find d
dx (cos(2x)).
d
dx (cos(2x)) = −2 sin(2x)
Step 3: Put the results from Step 1 and Step 2 together to differentiate
cos2(2x).
d
dx (cos2(2x)) = 2 cos(2x)·(−2 sin(2x)) = −4 cos(2x) sin(2x)
Step 4: Apply the same steps to differentiate sin2(3x).
d
dx (sin2(3x)) = 2 sin(3x)·3 cos(3x) = 6 sin(3x) cos(3x)
Step 5: Put the results from Step 3 and Step 4 together to find the derivative
of f(x).
f′(x) = −4 cos(2x) sin(2x) + 6 sin(3x) cos(3x)
Question 30
Question
Find the derivative of y= sin2(3x) + cos(2x) with respect to x.
18
Solution
Step 1: Use the chain rule to differentiate sin2(3x).
d
dx (sin2(3x)) = 2 sin(3x) cos(3x)·3 = 6 sin(3x) cos(3x)
Step 2: Differentiate cos(2x).
d
dx (cos(2x)) = −sin(2x)·2 = −2 sin(2x)
Step 3: The derivative of y= sin2(3x) + cos(2x) with respect to xis:
d
dx (y) = 6 sin(3x) cos(3x)−2 sin(2x)
Question 31
Question
Find the derivative of the function f(x) = sin2(3x) + cos3(2x).
Solution
Step 1: Apply the chain rule and power rule to differentiate sin2(3x).
f′(x) = 2 sin(3x)·cos(3x)·3
= 6 sin(3x) cos(3x).
Step 2: Apply the chain rule and power rule to differentiate cos3(2x).
f′(x) = −3 cos2(2x)·(−sin(2x)) ·2
= 6 cos2(2x) sin(2x).
Step 3: Add the derivatives of the two terms to find f′(x).
f′(x) = 6 sin(3x) cos(3x) + 6 cos2(2x) sin(2x).
Question 32
Question
Find the derivative of the function f(x) = sin(x) cos(x).
19
Solution
Step 1: Apply the product rule, (f g)′=f′g+fg′, where f(x) = sin(x) and
g(x) = cos(x). Step 2: Find f′(x) and g′(x). Step 3: Calculate the derivative of
f(x) = sin(x). Step 4: Use the derivative formula for sin(x), d
dx sin(x) = cos(x).
Step 5: Therefore, f′(x) = cos(x). Step 6: Calculate the derivative of g(x) =
cos(x). Step 7: Use the derivative formula for cos(x), d
dx cos(x) = −sin(x). Step
8: Thus, g′(x) = −sin(x). Step 9: Apply the product rule to find f(x)g′(x) +
f′(x)g(x). Step 10: Substitute f′(x) = cos(x) and g′(x) = −sin(x) into the
product rule. Step 11: Simplify the expression to get the final answer. Step 12:
The derivative of the function f(x) = sin(x) cos(x) is f′(x) = cos2(x)−sin2(x).
Question 33
Question
Find the derivative of y= sin2(5x3−2x) with respect to x.
Solution
Step 1: Apply chain rule - differentiate the outer function first, then the inner
function.
dy
dx = 2 sin5x3−2xcos5x3−2x(15x2−2)
= 2 sin5x3−2xcos5x3−2x(15x2−2)
Step 2: Simplify the expression by combining terms.
dy
dx = 30x2sin5x3−2xcos5x3−2x−4 sin5x3−2xcos5x3−2x
Step 3: Factor out sin5x3−2xcos5x3−2x.
dy
dx = (30x2−4) sin5x3−2xcos5x3−2x
Therefore, the derivative of y= sin2(5x3−2x) with respect to xis (30x2−
4) sin5x3−2xcos5x3−2x.
Question 34
Question
Find the derivative of the function f(x) = sin(x) cos(x).
20
Solution
To find the derivative of the function f(x) = sin(x) cos(x), we will use the
product rule of differentiation.
Step 1: Apply the product rule. Let u= sin(x) and v= cos(x).
f′(x) = u′v+uv′
= (sin′(x) cos(x) + sin(x) cos′(x))
= (cos(x) cos(x) + sin(x)(−sin(x)))
= cos2(x)−sin2(x)
= cos(2x).
Step 2: Simplify the final expression. The derivative of f(x) = sin(x) cos(x)
is f′(x) = cos(2x).
Question 35
Question
Find the derivative of the function f(x) = cos(3x) sin(4x).
Solution
Step 1: Apply the product rule, u′v+uv′, where u= cos(3x) and v= sin(4x).
Step 2: Find u′and v′.
u′=−3 sin(3x),
v′= 4 cos(4x).
Step 3: Apply the product rule formula.
f′(x) = u′v+uv′
= (−3 sin(3x))(sin(4x)) + (cos(3x))(4 cos(4x))
=−3 sin(3x) sin(4x) + 4 cos(3x) cos(4x).
Therefore, the derivative of the function f(x) = cos(3x) sin(4x) is f′(x) =
−3 sin(3x) sin(4x) + 4 cos(3x) cos(4x).
21
Question 2
Question
Find the derivative of the function f(x) = sin(x) cos(x).
Solution
Step 1: Apply the product rule, which states that if uand vare differentiable
functions of x, then the derivative of their product is (uv)′=u′v+uv′.
Step 2: Let u= sin(x) and v= cos(x).
Step 3: Find u′and v′.
Derivative of sin(x):
u′= cos(x)
Derivative of cos(x):
v′=−sin(x)
Step 4: Apply the product rule to find f′(x).
f′(x) = u′v+uv′
= (cos(x))(cos(x)) + (sin(x))(−sin(x))
= cos2(x)−sin2(x)
= cos(2x).
Therefore, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos(2x).
Question 3
Question
Find the derivative of the function f(x) = sin2(3x)−cos(2x) + tan(x).
Solution
Step 1: Apply the power rule and chain rule to find the derivative of sin2(3x).
f′(x) = d
dx (sin2(3x)) −
d
dx (cos(2x)) + d
dx (tan(x))
= 2 sin(3x) cos(3x)·3−(−sin(2x)·2) + sec2(x)
= 6 sin(3x) cos(3x) + 2 sin(2x) + sec2(x)
Therefore, the derivative of f(x) = sin2(3x)−cos(2x) + tan(x) is f′(x) =
6 sin(3x) cos(3x) + 2 sin(2x) + sec2(x).
2
Question 4
Question
Find the derivative of the function f(x) = sin(2x) cos(3x).
Solution
To find the derivative of f(x), we will use the product rule along with the chain
rule for differentiation.
Step 1: Apply the product rule. Let u= sin(2x) and v= cos(3x).
f′(x) = u′v+uv′
Step 2: Find u′and v′.
u′= 2 cos(2x), v′=−3 sin(3x)
Step 3: Substitute into the product rule formula and simplify.
f′(x) = (2 cos(2x)·cos(3x)) + (sin(2x)· −3 sin(3x))
f′(x) = 2 cos(2x) cos(3x)−3 sin(2x) sin(3x)
Therefore, the derivative of f(x) = sin(2x) cos(3x) is f′(x) = 2 cos(2x) cos(3x)−
3 sin(2x) sin(3x).
Question 5
Question
Find the derivative of f(x) = sin2(x) cos(x).
Solution
To find the derivative of f(x) = sin2(x) cos(x), we will use the product rule and
the chain rule.
Step 1: Apply the product rule. Let u= sin2(x) and v= cos(x). Then,
f′(x) = u′v+uv′.
Step 2: Find u′and v′. Differentiating u= sin2(x) with respect to x, we
get
u′= 2 sin(x) cos(x).
Differentiating v= cos(x) with respect to x, we get
v′=−sin(x).
3
Step 3: Substitute into the product rule formula. Substitute u,v,u′, and
v′into f′(x) = u′v+uv′to get
f′(x) = (2 sin(x) cos(x))(cos(x)) + (sin2(x))(−sin(x)).
Step 4: Simplify. Simplify the expression
f′(x) = 2 sin(x) cos2(x)−sin3(x).
Finally, the derivative of f(x) = sin2(x) cos(x) is f′(x) = 2 sin(x) cos2(x)−
sin3(x).
Question 6
Question
Find the derivative of f(x) = sin2(x) cos(x).
Solution
Step 1: Apply the product rule. Let u= sin2(x) and v= cos(x). Step 2: Find
u′and v′. Step 3: Apply the product rule to find f′(x). Step 4: Simplify the
expression for f′(x).
Step 1: Apply the product rule. Let u= sin2(x) and v= cos(x). Using the
product rule, we have
f′(x) = u′v+uv′.
Step 2: Find u′and v′. Taking the derivatives separately, we get
u′= 2 sin(x) cos(x)
and
v′=−sin(x).
Step 3: Apply the product rule to find f′(x). Substitute u,v,u′, and v′
into the product rule formula:
f′(x) = (2 sin(x) cos(x)) cos(x) + (sin2(x))(−sin(x)).
Step 4: Simplify the expression for f′(x). After simplifying, we get
f′(x) = 2 sin(x) cos2(x)−sin3(x).
Therefore, the derivative of f(x) = sin2(x) cos(x) is f′(x) = 2 sin(x) cos2(x)−
sin3(x).
Question 7
Question
Find the derivative of the function f(x) = sin(x)
1+cos(x).
4
Solution
Step 1: To find the derivative of f(x), we will use the quotient rule which
states that if u(x) and v(x) are differentiable functions, then d
dx u(x)
v(x)=
v(x)u′(x)−u(x)v′(x)
(v(x))2.
Step 2: Identify u(x) and v(x) in the function f(x). Here, u(x) = sin(x) and
v(x) = 1 + cos(x).
Step 3: Find u′(x) and v′(x) by taking the derivatives of u(x) and v(x).
u′(x) = d
dx sin(x) = cos(x),
v′(x) = d
dx (1 + cos(x)) = −sin(x).
Step 4: Apply the quotient rule to find f′(x).
f′(x) = (1 + cos(x)) cos(x)−sin(x)(−sin(x))
(1 + cos(x))2
=cos(x) + cos2(x) + sin2(x)
(1 + cos(x))2
=cos(x)+1
(1 + cos(x))2.
Therefore, the derivative of f(x) = sin(x)
1+cos(x)is f′(x) = cos(x)+1
(1+cos(x))2.
Question 8
Question
Find the derivative of the function f(x) = sin2(3x) + cos(2x).
Solution
Step 1: Apply the chain rule to differentiate sin2(3x).
d
dx [sin2(3x)] = 2 sin(3x) cos(3x)·3 = 6 sin(3x) cos(3x)
Step 2: Differentiate cos(2x).
d
dx [cos(2x)] = −sin(2x)·2 = −2 sin(2x)
Step 3: Combine the derivatives from Steps 1 and 2 to find the derivative
of f(x).
f′(x) = 6 sin(3x) cos(3x)−2 sin(2x)
5
Question 9
Question
Find the derivative of the function y=3 sin(x)−cos(x)
2 sin(x)+cos(x).
Solution
Step 1: To differentiate the given function, we will first rewrite it using the
quotient rule.
Step 2: The quotient rule states that for functions u(x) and v(x), the deriva-
tive of u(x)
v(x)is given by:
d
dx u(x)
v(x)=v(x)du
dx
−u(x)dv
dx
(v(x))2
Step 3: In this case, let u(x) = 3 sin(x)−cos(x) and v(x) = 2 sin(x)+cos(x).
Step 4: Calculate du
dx :
d
dx (3 sin(x)−cos(x)) = 3 cos(x) + sin(x)
Step 5: Calculate dv
dx :
d
dx (2 sin(x) + cos(x)) = 2 cos(x)−sin(x)
Step 6: Now, apply the quotient rule formula:
y′=(2 sin(x) + cos(x))(3 cos(x) + sin(x)) −(3 sin(x)−cos(x))(2 cos(x)−sin(x))
(2 sin(x) + cos(x))2
Step 7: Simplify the expression to find the derivative y′.
Step 8: After simplifying, the derivative of the function yis:
y′=
−5 sin(x)−2 cos(x)
(2 sin(x) + cos(x))2
Therefore, the derivative of the function y=3 sin(x)−cos(x)
2 sin(x)+cos(x)is y′=−5 sin(x)−2 cos(x)
(2 sin(x)+cos(x))2.
Question 10
Question
Find the derivative of the function f(x) = sin2(x)+cos2(x)
tan(x).
6
Solution
Step 1: Simplify the function.
f(x) = sin2(x) + cos2(x)
tan(x)
=1
tan(x)
= cot(x).
Step 2: Differentiate f(x) = cot(x) with respect to xusing the quotient rule.
d
dx (cot(x)) = d
dx cos(x)
sin(x)
=(sin(x)(−sin(x)) −cos(x) cos(x))
sin2(x)
=
−sin2(x)−cos2(x)
sin2(x)
=
−(sin2(x) + cos2(x))
sin2(x)
=
−1
sin2(x)
=−csc2(x).
Question 11
Question
Find the derivative of the function f(x) = sin2(3x)−cos2(2x).
Solution
Step 1: Apply the chain rule to differentiate sin2(3x). Step 2: Apply the chain
rule to differentiate cos2(2x). Step 3: Simplify the derivative of each term. Step
4: Combine the derivatives to find the derivative of the given function.
Step 1: Let u= 3x. Then, the derivative of sin2(3x) using the chain rule
is: d
dx (sin2(3x)) = 2 sin(3x)·cos(3x)·3 = 6 sin(3x) cos(3x)
Step 2: Let v= 2x. Then, the derivative of cos2(2x) using the chain rule
is: d
dx (cos2(2x)) = −2 cos(2x) sin(2x)·2 = −4 cos(2x) sin(2x)
7
Step 3: Simplify the derivatives we found in Step 1 and Step 2:
d
dx (sin2(3x)) = 6 sin(3x) cos(3x)
d
dx (cos2(2x)) = −4 cos(2x) sin(2x)
Step 4: Now, combine the derivatives to find the derivative of the given
function: d
dx (f(x)) = d
dx (sin2(3x)) −
d
dx (cos2(2x))
d
dx (f(x)) = 6 sin(3x) cos(3x)−(−4 cos(2x) sin(2x))
d
dx (f(x)) = 6 sin(3x) cos(3x) + 4 cos(2x) sin(2x)
Therefore, the derivative of the function f(x) = sin2(3x)−cos2(2x) is
6 sin(3x) cos(3x) + 4 cos(2x) sin(2x).
Question 12
Question
Find the derivative of the function f(x) = cos(x) sin(x).
Solution
Step 1: Apply the product rule to find the derivative of f(x). Step 2: Let
u(x) = cos(x) and v(x) = sin(x). Step 3: Find u′(x) and v′(x). Step 4: Apply
the product rule f′(x) = u(x)v′(x) + u′(x)v(x) to find f′(x). Step 5: Simplify
the result to obtain the final answer.
Question 13
Question
Find the derivative of f(x) = sin(2x) cos(3x).
Solution
Step 1: Use the product rule to differentiate f(x). Step 2: Apply the chain rule
to find the derivatives of sin(2x) and cos(3x). Step 3: Simplify the expression
by expanding and simplifying the trigonometric functions.
Step 1: Applying the product rule, we have:
f′(x) = (sin(2x))′(cos(3x)) + sin(2x)(cos(3x))′
8
Step 2: Now, let’s find the derivatives of sin(2x) and cos(3x) using the chain
rule:
(sin(2x))′= 2 cos(2x)
(cos(3x))′=−3 sin(3x)
Step 3: Substitute these derivatives back into the expression for f′(x):
f′(x) = (2 cos(2x))(cos(3x)) + sin(2x)(−3 sin(3x))
f′(x) = 2 cos(2x) cos(3x)−3 sin(2x) sin(3x)
Therefore, the derivative of f(x) = sin(2x) cos(3x) is f′(x) = 2 cos(2x) cos(3x)−
3 sin(2x) sin(3x).
Question 14
Question
Find the derivative of the function f(x) = tan(x) sin(2x).
Solution
Step 1: Apply the product rule to find the derivative of f(x). Step 2: Recall the
product rule states that if f(x) = g(x)h(x), then f′(x) = g′(x)h(x) + g(x)h′(x).
Step 1: Let g(x) = tan(x) and h(x) = sin(2x). Then, using the product rule,
f′(x) = g′(x)h(x) + g(x)h′(x)
Step 2: Find the derivatives g′(x) and h′(x). - To find g′(x), differentiate
tan(x) with respect to x. Differentiating tan(x), we have:
g′(x) = sec2(x)
- To find h′(x), differentiate sin(2x) with respect to xusing the chain rule.
Let u= 2x, then du
dx = 2. Substitute u= 2xinto sin(u) to get:
h′(x) = 2 cos(2x)
Substitute g′(x), h(x), g(x), and h′(x) into the product rule formula:
f′(x) = sec2(x) sin(2x) + tan(x)·(2 cos(2x))
Therefore, the derivative of f(x) = tan(x) sin(2x) is
f′(x) = sec2(x) sin(2x) + 2 tan(x) cos(2x)
.
9
Question 15
Question
Find the derivative of the function f(x) = cos2(3x) + sin2(4x).
Solution
Step 1: Recall the trigonometric identities cos2θ+ sin2θ= 1 and d
dx cos θ=
−sin θand d
dx sin θ= cos θ.
Step 2: Rewrite the function f(x) using the trigonometric identities:
f(x) = 1 + 1 = 2
Step 3: Find the derivative of f(x) using the constant multiple rule:
d
dx f(x) = d
dx 2=0
Step 4: Therefore, the derivative of the function f(x) = cos2(3x) + sin2(4x)
is 0 .
Question 16
Question
Find the derivative of the function f(x) = sin(x)−2 cos(x).
Solution
Step 1: Recall the derivatives of the trigonometric functions:
The derivative of sin(x) is cos(x).
The derivative of cos(x) is −sin(x).
Step 2: Find the derivative of f(x) = sin(x)−2 cos(x) using the sum/difference
rule for differentiation:
f′(x) = d
dx (sin(x)) −
d
dx (2 cos(x))
Step 3: Apply the derivatives of sin(x) and cos(x) that we recalled in Step
1:
f′(x) = cos(x)−2(−sin(x))
Step 4: Simplify the expression:
f′(x) = cos(x) + 2 sin(x)
Therefore, the derivative of the function f(x) = sin(x)−2 cos(x) is f′(x) =
cos(x) + 2 sin(x).
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Question 17
Question
Find the derivative of the function y= sin2(3x)−cos2(2x).
Solution
To find the derivative of the function y= sin2(3x)−cos2(2x), we will use the
chain rule and the derivative formulas for trigonometric functions.
Step 1: Find the derivative of sin2(3x).Let u= sin(3x). Then, the
derivative of sin2(3x) with respect to xcan be found using the chain rule:
d
dx (sin2(3x)) = 2 sin(3x) cos(3x)·3
= 6 sin(3x) cos(3x)
Step 2: Find the derivative of cos2(2x).Let v= cos(2x). Then, the
derivative of cos2(2x) with respect to xcan be found using the chain rule:
d
dx (cos2(2x)) = 2 cos(2x) sin(2x)·2
= 4 cos(2x) sin(2x)
Step 3: Combine the derivatives. Now, the derivative of y= sin2(3x)−
cos2(2x) is the sum of the derivatives we found in Step 1 and Step 2:
dy
dx = 6 sin(3x) cos(3x)−4 cos(2x) sin(2x)
Therefore, the derivative of y= sin2(3x)−cos2(2x) is 6 sin(3x) cos(3x)−
4 cos(2x) sin(2x).
Question 18
Question
Find the derivative of f(x) = sin(x)
cos(x).
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Solution
Step 1: Use the quotient rule to differentiate the function f(x) = sin(x)
cos(x).
f′(x) = d
dx sin(x)
cos(x)
=cos(x)·cos(x)−sin(x)·(−sin(x))
cos(x)2
=cos2(x) + sin2(x)
cos2(x)
=1
cos2(x)
= sec2(x)
So, the derivative of f(x) = sin(x)
cos(x)is f′(x) = sec2(x).
Question 19
Question
Find the derivative of y= sin2(3x) + cos(2x) with respect to x.
Solution
To find the derivative of y= sin2(3x) + cos(2x), we will use the chain rule
and derivative rules for trigonometric functions. Step 1: Find the derivative
of sin2(3x). Step 2: Find the derivative of cos(2x). Step 3: Combine the
derivatives to find dy
dx .
Step 1: Using the chain rule, let u= 3x:
d
dx (sin2(3x)) = d
du (sin2(u)) ·
du
dx
= 2 sin(3x) cos(3x)·3
= 6 sin(3x) cos(3x).
Step 2:
d
dx (cos(2x)) = −sin(2x)·2
=−2 sin(2x).
Step 3: Putting the derivatives together:
dy
dx = 6 sin(3x) cos(3x)−2 sin(2x)
= 3 sin(6x)−2 sin(2x).
So, the derivative of y= sin2(3x) + cos(2x) with respect to xis dy
dx =
3 sin(6x)−2 sin(2x).
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Question 20
Question
Find the derivative of y= sin(2x) cos(3x) with respect to x.
Solution
Step 1: Apply the product rule, which states that if uand vare functions of x,
then the derivative of their product is u′v+uv′.
Let u= sin(2x) and v= cos(3x). Then u′= 2 cos(2x) and v′=−3 sin(3x).
Step 2: Compute the derivative using the product rule as follows:
d
dx [sin(2x) cos(3x)] = u′v+uv′
= (2 cos(2x))(cos(3x)) + (sin(2x))(−3 sin(3x))
= 2 cos(2x) cos(3x)−3 sin(2x) sin(3x).
Therefore, the derivative of y= sin(2x) cos(3x) with respect to xis 2 cos(2x) cos(3x)−
3 sin(2x) sin(3x).
Question 21
Question
Find the derivative of the function f(x) = sin2(x)−cos2(x)
tan(x).
Solution
To find the derivative of the function f(x), we will first simplify the function
using trigonometric identities, then apply the quotient rule and chain rule to
find the derivative.
Step 1: Simplify the function using trigonometric identities.
f(x) = sin2(x)−cos2(x)
tan(x)
=sin2(x)−cos2(x)
sin(x)
cos(x)
= (sin2(x)−cos2(x)) ·
cos(x)
sin(x)
= sin(x) cos(x)−cos(x)
= sin(x)(cos(x)−1)
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Step 2: Use the product rule and chain rule to find the derivative of f(x).
Let u= sin(x) and v= cos(x)−1.
d
dx [sin(x)(cos(x)−1)] = du
dx
·v+u·
dv
dx
= (cos(x))(cos(x)−1) + sin(x)(−sin(x))
= cos2(x)−cos(x)−sin2(x)
Step 3: Simplify the result. Recall that cos2(x) + sin2(x) = 1. Therefore,
the derivative is:
cos2(x)−cos(x)−sin2(x) = 1 −cos(x)−1 = −cos(x)
Therefore, the derivative of the function f(x) is −cos(x).
Question 22
Question
Find the derivative of the function f(x) = sin2(x) cos(x).
Solution
Step 1: Use the product rule to differentiate f(x) = sin2(x) cos(x). Step 2: Let
u= sin2(x) and v= cos(x). Step 3: Find u′and v′.
u′=d
dx (sin2(x))
= 2 sin(x) cos(x)
= 2 sin(x) cos(x)
v′=d
dx (cos(x))
=−sin(x)
Step 4: Apply the product rule f′(x) = u′v+uv′.
f′(x) = (2 sin(x) cos(x)) cos(x) + sin2(x)(−sin(x))
= 2 sin(x) cos2(x)−sin3(x)
Therefore, the derivative of f(x) = sin2(x) cos(x) is f′(x) = 2 sin(x) cos2(x)−
sin3(x).
Question 23
Question
Find the derivative of f(x) = sin(x) cos(x).
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Solution
To find the derivative of f(x) = sin(x) cos(x), we will use the product rule,
which states that if h(x) = g(x)·j(x), then h′(x) = g′(x)·j(x) + g(x)·j′(x).
Step 1: Let g(x) = sin(x) and j(x) = cos(x). Then, we have g′(x) = cos(x)
and j′(x) = −sin(x).
Step 2: Apply the product rule to find f′(x):
f′(x) = g′(x)·j(x) + g(x)·j′(x)
= cos(x)·cos(x) + sin(x)·(−sin(x))
= cos2(x)−sin2(x)
= cos(2x).
Therefore, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos(2x).
Question 24
Question
Find the derivative of the function f(x) = sin(x) cos(x).
Solution
Step 1: Apply the product rule, d
dx (u·v) = u′
·v+u·v′, where u= sin(x) and
v= cos(x). Step 2: Calculate the derivatives of uand v. Step 3: Let’s find u′
and v′. Step 4:
u′=d
dx (sin(x)) = cos(x)
v′=d
dx (cos(x)) = −sin(x)
Step 5: Substitute u′,v′,u, and vinto the product rule formula. Step 6: So,
d
dx (sin(x) cos(x)) = cos(x) cos(x) + sin(x)·(−sin(x))
= cos2(x)−sin2(x)
Therefore, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos2(x)−sin2(x).
Question 25
Question
Differentiate the function f(x) = sin2(2x) + sin2(x) with respect to x.
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Solution
Step 1: Apply the chain rule to differentiate sin2(2x). Step 2: Apply the chain
rule to differentiate sin2(x). Step 3: Combine the results to find the derivative
of f(x).
Step 1: Let u= sin(2x), then f(x) = u2. Using the chain rule, we have
df
dx =d
dx (sin(2x))2= 2 sin(2x) cos(2x)·2 = 4 sin(2x) cos(2x).
Step 2: Let v= sin(x), then g(x) = v2. Applying the chain rule, we get
dg
dx =d
dx (sin(x))2= 2 sin(x) cos(x) = 2 sin(x) cos(x).
Step 3: Now, differentiate the function f(x) = sin2(2x) + sin2(x):
f′(x) = 4 sin(2x) cos(2x) + 2 sin(x) cos(x).
Therefore, the derivative of f(x) with respect to xis f′(x) = 4 sin(2x) cos(2x)+
2 sin(x) cos(x).
Question 26
Question
Compute the derivative of f(x) = tan(x)·cos(x).
Solution
Step 1: We will use the product rule to differentiate f(x), which states that if
f(x) = u(x)·v(x), then f′(x) = u(x)·v′(x) + u′(x)·v(x).
Step 2: Let u(x) = tan(x) and v(x) = cos(x).
Step 3: Find u′(x) and v′(x).
u′(x) = d
dx (tan(x)) = sec2(x) by using the derivative of tan(x).
v′(x) = d
dx (cos(x)) = −sin(x) by using the derivative of cos(x).
Step 4: Apply the product rule to find f′(x).
f′(x) = tan(x)·(−sin(x)) + sec2(x)·cos(x)
Step 5: Simplify the expression.
f′(x) = −tan(x)·sin(x) + sec2(x)·cos(x)
Step 6: Thus, the derivative of f(x) = tan(x)·cos(x) is f′(x) = −tan(x)·
sin(x) + sec2(x)·cos(x).
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Question 27
Question
Find the derivative of the function f(x) = sin(2x)+3 cos(x)
x.
Solution
Step 1: To find the derivative of f(x), we will use the quotient rule. The quotient
rule states that if f(x) = g(x)
h(x), then f′(x) = g′(x)h(x)−g(x)h′(x)
(h(x))2.
Step 2: Let g(x) = sin(2x) + 3 cos(x) and h(x) = x. We can find the
derivatives of g(x) and h(x):
For g(x):
g′(x) = d
dx (sin(2x) + 3 cos(x))
g′(x) = 2 cos(2x)−3 sin(x)
For h(x):
h′(x)=1
Step 3: Now, we can apply the quotient rule to find f′(x):
f′(x) = (2 cos(2x)−3 sin(x))(x)−(sin(2x) + 3 cos(x))(1)
x2
f′(x) = 2xcos(2x)−3xsin(x)−sin(2x)−3 cos(x)
x2
Therefore, the derivative of the function f(x) = sin(2x)+3 cos(x)
xis f′(x) =
2xcos(2x)−3xsin(x)−sin(2x)−3 cos(x)
x2.
Question 28
Question
Compute the derivative of f(x) = 3 sin(2x) + 4 cos(3x).
Solution
Step 1: Apply the derivative formulas for sine and cosine functions.
Step 2: Compute the derivative of f(x) using the chain rule.
Step 3: Simplify the final result.
Step 1:
The derivative of sin(ax) is acos(ax) and the derivative of cos(ax) is −asin(ax).
Therefore, the derivative of f(x) is:
f′(x) = 3(2 cos(2x)) + 4(−3 sin(3x))
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Step 2:
Applying the chain rule, we have:
f′(x) = 6 cos(2x)−12 sin(3x)
Step 3:
Simplify the final result:
f′(x) = 6 cos(2x)−12 sin(3x)
Question 29
Question
Find the derivative of the function f(x) = cos2(2x) + sin2(3x).
Solution
Step 1: Apply the chain rule to differentiate cos2(2x).
d
dx (cos2(2x)) = 2 cos(2x)·
d
dx (cos(2x))
Step 2: Apply the chain rule and power rule to find d
dx (cos(2x)).
d
dx (cos(2x)) = −2 sin(2x)
Step 3: Put the results from Step 1 and Step 2 together to differentiate
cos2(2x).
d
dx (cos2(2x)) = 2 cos(2x)·(−2 sin(2x)) = −4 cos(2x) sin(2x)
Step 4: Apply the same steps to differentiate sin2(3x).
d
dx (sin2(3x)) = 2 sin(3x)·3 cos(3x) = 6 sin(3x) cos(3x)
Step 5: Put the results from Step 3 and Step 4 together to find the derivative
of f(x).
f′(x) = −4 cos(2x) sin(2x) + 6 sin(3x) cos(3x)
Question 30
Question
Find the derivative of y= sin2(3x) + cos(2x) with respect to x.
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Solution
Step 1: Use the chain rule to differentiate sin2(3x).
d
dx (sin2(3x)) = 2 sin(3x) cos(3x)·3 = 6 sin(3x) cos(3x)
Step 2: Differentiate cos(2x).
d
dx (cos(2x)) = −sin(2x)·2 = −2 sin(2x)
Step 3: The derivative of y= sin2(3x) + cos(2x) with respect to xis:
d
dx (y) = 6 sin(3x) cos(3x)−2 sin(2x)
Question 31
Question
Find the derivative of the function f(x) = sin2(3x) + cos3(2x).
Solution
Step 1: Apply the chain rule and power rule to differentiate sin2(3x).
f′(x) = 2 sin(3x)·cos(3x)·3
= 6 sin(3x) cos(3x).
Step 2: Apply the chain rule and power rule to differentiate cos3(2x).
f′(x) = −3 cos2(2x)·(−sin(2x)) ·2
= 6 cos2(2x) sin(2x).
Step 3: Add the derivatives of the two terms to find f′(x).
f′(x) = 6 sin(3x) cos(3x) + 6 cos2(2x) sin(2x).
Question 32
Question
Find the derivative of the function f(x) = sin(x) cos(x).
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Solution
Step 1: Apply the product rule, (f g)′=f′g+fg′, where f(x) = sin(x) and
g(x) = cos(x). Step 2: Find f′(x) and g′(x). Step 3: Calculate the derivative of
f(x) = sin(x). Step 4: Use the derivative formula for sin(x), d
dx sin(x) = cos(x).
Step 5: Therefore, f′(x) = cos(x). Step 6: Calculate the derivative of g(x) =
cos(x). Step 7: Use the derivative formula for cos(x), d
dx cos(x) = −sin(x). Step
8: Thus, g′(x) = −sin(x). Step 9: Apply the product rule to find f(x)g′(x) +
f′(x)g(x). Step 10: Substitute f′(x) = cos(x) and g′(x) = −sin(x) into the
product rule. Step 11: Simplify the expression to get the final answer. Step 12:
The derivative of the function f(x) = sin(x) cos(x) is f′(x) = cos2(x)−sin2(x).
Question 33
Question
Find the derivative of y= sin2(5x3−2x) with respect to x.
Solution
Step 1: Apply chain rule - differentiate the outer function first, then the inner
function.
dy
dx = 2 sin5x3−2xcos5x3−2x(15x2−2)
= 2 sin5x3−2xcos5x3−2x(15x2−2)
Step 2: Simplify the expression by combining terms.
dy
dx = 30x2sin5x3−2xcos5x3−2x−4 sin5x3−2xcos5x3−2x
Step 3: Factor out sin5x3−2xcos5x3−2x.
dy
dx = (30x2−4) sin5x3−2xcos5x3−2x
Therefore, the derivative of y= sin2(5x3−2x) with respect to xis (30x2−
4) sin5x3−2xcos5x3−2x.
Question 34
Question
Find the derivative of the function f(x) = sin(x) cos(x).
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Solution
To find the derivative of the function f(x) = sin(x) cos(x), we will use the
product rule of differentiation.
Step 1: Apply the product rule. Let u= sin(x) and v= cos(x).
f′(x) = u′v+uv′
= (sin′(x) cos(x) + sin(x) cos′(x))
= (cos(x) cos(x) + sin(x)(−sin(x)))
= cos2(x)−sin2(x)
= cos(2x).
Step 2: Simplify the final expression. The derivative of f(x) = sin(x) cos(x)
is f′(x) = cos(2x).
Question 35
Question
Find the derivative of the function f(x) = cos(3x) sin(4x).
Solution
Step 1: Apply the product rule, u′v+uv′, where u= cos(3x) and v= sin(4x).
Step 2: Find u′and v′.
u′=−3 sin(3x),
v′= 4 cos(4x).
Step 3: Apply the product rule formula.
f′(x) = u′v+uv′
= (−3 sin(3x))(sin(4x)) + (cos(3x))(4 cos(4x))
=−3 sin(3x) sin(4x) + 4 cos(3x) cos(4x).
Therefore, the derivative of the function f(x) = cos(3x) sin(4x) is f′(x) =
−3 sin(3x) sin(4x) + 4 cos(3x) cos(4x).
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