MATH 108 - ELEMENTARY AND INTERMEDIATE ALGEBRA Differentiation of Trigonometric Functions Question Bank Set 3 | MATH 108

MATH 108 - ELEMENTARY AND

INTERMEDIATE ALGEBRA -

Differentiation of Trigonometric

Functions

Question Bank - Set 3

Liberty University

Question 1

Question

Find the derivative of the function f(x) = cos(2x) sin(3x).

Solution

Step 1: Use the product rule to find the derivative of f(x). Step 2: Apply the

chain rule to differentiate cos(2x) and sin(3x). Step 3: Simplify the result to

get the final derivative.

Step 1: Apply the product rule to find f′(x). Let u(x) = cos(2x) and

v(x) = sin(3x). Then, f(x) = u(x)v(x). By the product rule, f′(x) = u′v+uv′.

Step 2: Differentiate cos(2x) and sin(3x) using the chain rule.

u′(x) = −sin(2x)·2 and v′(x) = cos(3x)·3

Step 3: Substitute u′(x), v′(x), u(x), and v(x) into the formula for f′(x).

f′(x)=(−sin(2x)·2) sin(3x) + cos(2x)(cos(3x)·3)

f′(x) = −2 sin(2x) sin(3x) + 3 cos(2x) cos(3x)

Therefore, the derivative of f(x) = cos(2x) sin(3x) is f′(x) = −2 sin(2x) sin(3x)+

3 cos(2x) cos(3x).

Question 2

Question

Find the derivative of the function f(x) = sin2(3x)−cos2(2x).

Solution

Step 1: Apply the chain rule to differentiate sin2(3x).

Let u= sin(3x) =⇒du

dx = 3 cos(3x)

Now, differentiate sin2(3x) = u2with respect to x:d

dx(sin2(3x)) = 2 sin(3x)·3 cos(3x)

= 6 sin(3x) cos(3x)

Step 2: Apply the chain rule to differentiate cos2(2x).

Let v= cos(2x) =⇒dv

dx =−2 sin(2x)

Now, differentiate cos2(2x) = v2with respect to x:d

dx(cos2(2x)) = 2 cos(2x)·−2 sin(2x)

=−4 cos(2x) sin(2x)

Step 3: Combine the derivatives of sin2(3x) and cos2(2x) to find the deriva-

tive of f(x).

f′(x) = 6 sin(3x) cos(3x)+(−4 cos(2x) sin(2x))

f′(x) = 2 sin(3x) cos(3x)−4 cos(2x) sin(2x)

Therefore, the derivative of f(x) = sin2(3x)−cos2(2x) is f′(x) = 2 sin(3x) cos(3x)−

4 cos(2x) sin(2x).

Question 3

Question

Find the derivative of the function f(x) = cos2(3x).

Solution

Step 1: Apply the chain rule, which states that if uand vare both differentiable

functions of x, then the derivative of u(v(x)) with respect to xis u′(v(x))·v′(x).

Step 2: Let u= cos(x) and v= 3x. Then f(x) = u2(v) = u(v(x))2=

(cos(3x))2.

Step 3: Find f′(x) using the chain rule and the derivative of cos(x).

f′(x) = 2 cos(3x)·(cos(3x))′

= 2 cos(3x)·(−3 sin(3x))

=−6 cos(3x) sin(3x)

Therefore, the derivative of f(x) = cos2(3x) is f′(x) = −6 cos(3x) sin(3x).

Question 4

Question

Find the derivative of the function f(x) = sin(2x) cos(x).

Solution

Step 1: Apply the product rule to find the derivative of f(x). Step 2: Recall

that the product rule states that the derivative of the product of two functions

is given by (fg)′=f′g+f g′. Step 3: Let f(x) = sin(2x) and g(x) = cos(x).

Step 4: Find f′(x) and g′(x). Step 5: The derivative of sin(2x) with respect to

xis d

dx (sin(2x)) = 2 cos(2x) by the chain rule. Step 6: The derivative of cos(x)

with respect to xis d

dx (cos(x)) = −sin(x). Step 7: Now, apply the product rule

to find f′(x) and g′(x). Step 8: f′(x) = cos(2x)·2 by the chain rule. Step 9:

g′(x) = −sin(x). Step 10: Now, put all the pieces together using the product

rule: f′(x)g(x) + f(x)g′(x). Step 11: So, the derivative of f(x) = sin(2x) cos(x)

is 2 cos(2x) cos(x)−sin(2x) sin(x).

Question 5

Question

Find the derivative of the function f(x) = sin3(2x) + cos2(x).

Solution

Step 1: To find the derivative of f(x), we will differentiate each term separately

using the rules of differentiation.

Step 2: Let’s start by finding the derivative of sin3(2x). Using the chain

rule, the derivative of sin3(2x) will be:

dx sin3(2x) = 3 sin2(2x) cos(2x)·2 = 6 sin2(2x) cos(2x).

Step 3: Next, let’s find the derivative of cos2(x). The derivative of cos2(x)

will be: d

dx cos2(x) = 2 cos(x)(−sin(x)) = −2 cos(x) sin(x).

Step 4: Therefore, the derivative of f(x) = sin3(2x) + cos2(x) will be:

f′(x) = 6 sin2(2x) cos(2x)−2 cos(x) sin(x).

So, the derivative of f(x) is f′(x) = 6 sin2(2x) cos(2x)−2 cos(x) sin(x).

Question 6

Question

Compute the derivative of f(x) = sin2(3x) + cos2(4x).

Solution

Step 1: Use the trigonometric identity sin2(θ) + cos2(θ) = 1 to simplify f(x).

f(x) = sin2(3x) + cos2(4x)

= 1

Step 2: Since f(x) = 1 for all x, the derivative of f(x) is 0.

f′(x)=0

Question 7

Question

Find the derivative of the function f(x) = cos(3x) sin(4x).

Solution

Step 1: Apply the product rule to differentiate the function. Step 2: Use the

chain rule to differentiate the trigonometric functions. Step 3: Simplify the

expression to get the final derivative.

Step 1:

Apply the product rule, which states that (uv)′=u′v+uv′, where uand vare

functions of x. Let u(x) = cos(3x) and v(x) = sin(4x). Then, the derivative of

f(x) is given by:

f′(x) = u′(x)v(x) + u(x)v′(x)

Step 2:

Now, differentiate u(x) = cos(3x) and v(x) = sin(4x). We have:

u′(x) = −3 sin(3x)

v′(x) = 4 cos(4x)

Step 3:

Now, substitute these derivatives back into the product rule formula:

f′(x)=(−3 sin(3x)) sin(4x) + cos(3x)(4 cos(4x))

f′(x) = −3 sin(3x) sin(4x) + 4 cos(3x) cos(4x)

f′(x) = −sin(7x) + cos(x)

Therefore, the derivative of the function f(x) = cos(3x) sin(4x) is f′(x) =

−sin(7x) + 4 cos(3x) cos(4x).

Question 8

Question

Find the derivative of f(x) = sinx2cos(2x).

Solution

Step 1: We will use the product rule to differentiate the given function. 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x2and h(x) = cos(2x).

Step 3: Calculate g′(x) and h′(x).

g′(x) = (sinx2)′

= cosx2·(x2)′

= cosx2·2x

= 2xcosx2

Step 4:

h′(x) = (cos(2x))′

=−sin(2x)·(2x)′

=−2 sin(2x)

Step 5: Apply the product rule:

f′(x) = (2xcosx2)(cos(2x)) + (sinx2)(−2 sin(2x))

= 2xcosx2cos(2x)−2 sinx2sin(2x)

Therefore, the derivative of f(x) = sinx2cos(2x) is f′(x) = 2xcosx2cos(2x)−

2 sinx2sin(2x).

Question 9

Question

Find the derivative of f(x) = sin(x) cos(x).

Solution

Step 1: Use the product rule to differentiate f(x) = sin(x) cos(x). Step 2: Let

u= sin(x) and v= cos(x). Step 3: Find u′and v′. Step 4: Apply the product

rule: f′(x) = u′v+uv′. Step 5: Substitute the values of u,v,u′, and v′back

into the formula to find the derivative of f(x).

Let’s start solving the problem:

1. Step 1: Apply the product rule:

f′(x) = (sin(x))′(cos(x)) + sin(x)(cos(x))′.

2. Step 2: Let u= sin(x) and v= cos(x).

3. Step 3: Find u′and v′:

u′= cos(x) and v′=−sin(x).

4. Step 4: Apply the product rule:

f′(x) = (cos(x) cos(x)) + (sin(x)(−sin(x))).

5. Step 5: Calculate the derivatives and simplify:

f′(x) = cos2(x)−sin2(x).

Therefore, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos2(x)−

sin2(x).

Question 10

Question

Find the derivative of y= sin2(3x) + cos2(2x) with respect to x.

Solution

Step 1: Apply the chain rule for sin2(3x).

Step 2: Apply the chain rule for cos2(2x).

Step 3: Combine the derivatives of both terms to find the overall derivative

of y.

Step 1: We have y= sin2(3x), so let u= 3x. Thus, we can rewrite yas

y= (sin u)2. Applying the chain rule gives:

dx = 2 sin(3x) cos(3x)·3 = 6 sin(3x) cos(3x)

Step 2: We have y= cos2(2x), so let v= 2x. Thus, we can rewrite yas

y= (cos v)2. Applying the chain rule gives:

dx = 2 cos(2x) sin(2x)·2 = 4 cos(2x) sin(2x)

Step 3: Combining the derivatives of both terms gives:

dx = 6 sin(3x) cos(3x) + 4 cos(2x) sin(2x) = 6 sin(6x) + 4 sin(4x)

Question 11

Question

Calculate the derivative of the function f(x) = sin(x) cos(x).

Solution

To find the derivative of the function f(x) = sin(x) cos(x), we will use the

product rule.

Step 1: Apply the product rule. Let u= sin(x) and v= cos(x). Then,

u′= cos(x) and v′=−sin(x).

Step 2: Use the product rule formula: (uv)′=u′v+uv′.

f′(x) = (sin(x) cos(x))′

= sin(x)·(−sin(x)) + cos(x)·cos(x)

=−sin2(x) + cos2(x)

Step 3: Use the Pythagorean identity sin2(x) + cos2(x) = 1.

f′(x) = −(1) = −1

Therefore, the derivative of f(x) = sin(x) cos(x) is −1.

Question 12

Question

Determine the derivative of f(x) = cos2(x) tan(x).

Solution

To find the derivative of f(x), we will use the product rule and the chain rule

for differentiation.

f(x) = cos2(x) tan(x)

= (cos(x) cos(x))(sin(x)/cos(x))

= cos(x) cos(x)·sin(x)

cos(x)

= cos(x)·cos(x)·sin(x)

cos(x)

= cos(x)·sin(x)

= sin(x) cos(x)

Now, we can differentiate f(x):

f′(x) = (sin(x))′·(cos(x)) + (cos(x))′·(sin(x))

= cos(x) cos(x)+(−sin(x)) sin(x)

= cos2(x)−sin2(x)

Therefore, the derivative of f(x) = cos2(x) tan(x) is f′(x) = cos2(x)−

sin2(x).

Question 13

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 f(x). Step 2: Use the

chain rule to differentiate sin(2x) and cos(3x). Step 3: Simplify the expression

and combine like terms to get the final answer.

Step 1:

Apply the product rule:

f′(x) = d

dx (sin(2x) cos(3x)) = sin(2x)d

dx cos(3x) + cos(3x)d

dx sin(2x)

Step 2:

Differentiate sin(2x) and cos(3x) using the chain rule:

dx sin(2x) = cos(2x)·2

dx cos(3x) = −sin(3x)·3

Step 3:

Substitute the derivatives back into the expression:

f′(x) = sin(2x)(−3 sin(3x)) + cos(3x)(2 cos(2x))

f′(x) = −3 sin(2x) sin(3x) + 2 cos(3x) cos(2x)

Therefore, the derivative of f(x) = sin(2x) cos(3x) is f′(x) = −3 sin(2x) sin(3x)+

2 cos(3x) cos(2x).

Question 14

Question

Find the derivative of the function f(x) = sin2(2x) + cos2(3x).

Solution

Step 1: Apply the chain rule to differentiate sin2(2x).

dx(sin2(2x)) = 2 sin(2x) cos(2x)

Step 2: Apply the chain rule to differentiate cos2(3x).

dx(cos2(3x)) = −2 cos(3x) sin(3x)

Step 3: Combine the derivatives of each term to find the derivative of the

whole function f(x).

f′(x) = 2 sin(2x) cos(2x)−2 cos(3x) sin(3x)

Therefore, the derivative of f(x) = sin2(2x)+cos2(3x) is f′(x) = 2 sin(2x) cos(2x)−2 cos(3x) sin(3x) .

Question 15

Question

Find the derivative of the function f(x) = tan(x)−sin(x).

Solution

Step 1: Recall the derivatives of the trigonometric functions:



dx (sin(x)) = cos(x)



dx (cos(x)) = −sin(x)



dx (tan(x)) = sec2(x)

Step 2: Use the derivative rules to find f′(x):

f′(x) = d

dx(tan(x)) −d

dx(sin(x))

Step 3: Substitute the derivatives of tan(x) and sin(x):

f′(x) = sec2(x)−cos(x)

So, the derivative of f(x) = tan(x)−sin(x) is f′(x) = sec2(x)−cos(x).

Question 16

Question

Find the derivative of the function f(x) = sin2(3x)−cos2(2x).

Solution

To find the derivative f′(x) of the given function f(x), we will use the chain

rule and the power rule for differentiation.

Step 1: Find the derivative of sin2(3x). Let u= sin(3x). Then, u2=

sin2(3x). Using the chain rule, we have:

dx[sin2(3x)] = d

du(u2)·du

dx = 2ucos(3x)·3.

Thus, the derivative of sin2(3x) is 2 sin(3x) cos(3x)·3.

Step 2: Find the derivative of cos2(2x). Let v= cos(2x). Then, v2=

cos2(2x). Using the chain rule, we have:

dx[cos2(2x)] = d

dv (v2)·dv

dx = 2v(−sin(2x)) ·2.

Thus, the derivative of cos2(2x) is −4 cos(2x) sin(2x).

Step 3: Combine the derivatives. The derivative of the given function

f(x) = sin2(3x)−cos2(2x) is:

f′(x) = 2 sin(3x) cos(3x)·3+(−4 cos(2x) sin(2x)).

So, the derivative of f(x) is 6 sin(3x) cos(3x)−4 cos(2x) sin(2x).

Question 17

Question

Find the derivative of y=sin(2x)

cos(x).

Solution

To find the derivative of y=sin(2x)

cos(x), we will use the quotient rule and the chain

rule of differentiation.

Step 1: Apply the quotient rule, which states that if uand vare differen-

tiable functions, then u

v′=u′v−uv′

v2. Let u= sin(2x) and v= cos(x). Then,

u′= 2 cos(2x) and v′=−sin(x).

Step 2: Compute y′using the quotient rule.

y′=(2 cos(2x)·cos(x)−sin(2x)· − sin(x))

cos2(x)

=2 cos(2x) cos(x) + sin(2x) sin(x)

cos2(x)

Step 3: Apply the angle sum identities cos(a+b) = cos(a) cos(b)−sin(a) sin(b)

and sin(a+b) = sin(a) cos(b) + cos(a) sin(b) to simplify the expression.

y′=2(cos(x) cos(2x) + sin(x) sin(2x))

cos2(x)

Step 4: Use the angle sum identities again to further simplify.

y′=2 cos(x+ 2x)

cos2(x)=2 cos(3x)

cos2(x)

Therefore, the derivative of y=sin(2x)

cos(x)is y′=2 cos(3x)

cos2(x).

Question 18

Question

Find the derivative of the function f(x) = cos(2x) sin(3x).

Solution

Step 1: Apply the product rule to find the derivative of f(x). Step 2: Recall

that the product rule states (u·v)′=u′v+uv′. In this case, let u= cos(2x) and

v= sin(3x). Step 3: Find u′and v′. Step 4: u′=−2 sin(2x) and v′= 3 cos(3x).

Step 5: Now apply the product rule to find f′(x). Step 6: f′(x) = cos(2x)·

3 cos(3x)+(−2 sin(2x))·sin(3x). Step 7: Simplify the expression to get the final

answer. Step 8: f′(x) = 3 cos(2x) cos(3x)−2 sin(2x) sin(3x).

Question 19

Question

Find the derivative of the function f(x) = sin(2x) cos(3x).

Solution

To find the derivative of f(x) = sin(2x) cos(3x), we will use the product rule.

Step 1: Apply the product rule. If f(x) = g(x)·h(x), then f′(x) = g′(x)·

h(x) + g(x)·h′(x).

Let g(x) = sin(2x) and h(x) = cos(3x).

Step 2: Find g′(x) and h′(x).



Calculate g′(x) by applying the chain rule: g′(x) = 2 cos(2x).



Calculate h′(x) by applying the chain rule: h′(x) = −3 sin(3x).

Step 3: Substitute g(x), g′(x), h(x), and h′(x) back into the product rule

formula.

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 f(x) = sin(2x) cos(3x) is f′(x) = 2 cos(2x) cos(3x)−

3 sin(2x) sin(3x).

Question 20

Question

Find the derivative of y=cos(x) sin(x)

Solution

Step 1: To find the derivative using the quotient rule, let u(x) = cos(x) sin(x)

and v(x) = x.

Step 2: Find u′(x).

u′(x) = (cos(x))′(sin(x)) + cos(x)(sin(x))′= (−sin(x)) sin(x) + cos(x) cos(x)

u′(x) = −sin2(x) + cos2(x)

Step 3: Find v′(x).

v′(x)=1

Step 4: Apply the quotient rule.

dx u(x)

v(x)=u′(x)v(x)−u(x)v′(x)

(v(x))2

dx cos(x) sin(x)

x=(−sin2(x) + cos2(x))x−(cos(x) sin(x))(1)

Step 5: Simplify the expression.

dx cos(x) sin(x)

x=−xsin2(x) + xcos2(x)−cos(x) sin(x)

dx cos(x) sin(x)

x=x(cos2(x)−sin2(x)) −cos(x) sin(x)

Hence, the derivative of y=cos(x) sin(x)

xis y′=x(cos2(x)−sin2(x))−cos(x) sin(x)

x2.

Question 21

Question

Find the derivative of y= sin(2x) cos(3x).

Solution

Step 1: Apply the product rule to differentiate the given function.

Step 2: Let u= sin(2x) and v= cos(3x).

Step 3: Find u′and v′.

u′=d

dx(sin(2x)) = 2 cos(2x)

v′=d

dx(cos(3x)) = −3 sin(3x)

Step 4: Apply the product rule to get y′.

y′=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) is 2 cos(2x) cos(3x)−3 sin(2x) sin(3x).

Question 22

Question

Find the derivative of the function f(x) = sin(3x) cos(2x).

Solution

Step 1: Apply the product rule. Step 2: Recall 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(3x)

and v(x) = cos(2x). Step 4: Find u′(x) and v′(x). Step 5: u′(x) = 3 cos(3x) by

applying the chain rule. Step 6: v′(x) = −2 sin(2x) by applying the chain rule.

Step 7: Substitute u(x), v(x), u′(x), and v′(x) into the product rule formula.

Step 8: f′(x) = (3 cos(3x) cos(2x))+(sin(3x)(−2 sin(2x))). Step 9: Simplify the

expression. Step 10: f′(x) = 3 cos(3x) cos(2x)−2 sin(3x) sin(2x).

Question 23

Question

Find the derivative of the function f(x) = cos2(x) + sin(2x).

Solution

Step 1: We will differentiate each term in the function f(x) separately using the

rules of differentiation.

Step 2: For the first term, cos2(x), we will use the chain rule. Let u= cos(x).

Then, f(x) = u2and df

dx = 2udu

dx .

Step 3: Differentiating cos(x) with respect to xgives d

dx (cos(x)) = −sin(x).

Step 4: Applying the chain rule, the derivative of cos2(x) is d

dx (cos2(x)) =

2 cos(x)(−sin(x)).

Step 5: Simplifying, we have d

dx (cos2(x)) = −2 cos(x) sin(x).

Step 6: For the second term sin(2x), we can use the trigonometric identity

sin(2x) = 2 sin(x) cos(x).

Step 7: So, the derivative of sin(2x) is d

dx (sin(2x)) = d

dx (2 sin(x) cos(x)) =

2(cos(x) cos(x)−sin(x) sin(x)).

Step 8: Simplifying, we get d

dx (sin(2x)) = 2(cos2(x)−sin2(x)).

Step 9: Therefore, the derivative of the function f(x) = cos2(x) + sin(2x) is:

dx(f(x)) = −2 cos(x) sin(x) + 2(cos2(x)−sin2(x))

Question 24

Question

Differentiate the function f(x) = cos2(3x) with respect to x.

Solution

Step 1: Apply the chain rule by letting u= 3x. Step 2: Calculate df

du . Step

3: Calculate du

dx . Step 4: Substitute the results from Step 2 and Step 3 into

the chain rule formula. Step 5: Simplify the expression to find the derivative of

f(x).

Step 1: Let u= 3x, so f(x) = cos2(u).

Step 2: Calculate df

du by differentiating cos2(u) with respect to u:

du = 2 cos(u)·(−sin(u)) = −2 cos(u) sin(u)

Step 3: Calculate du

dx :

dx = 3

Step 4: Apply the chain rule:

dx =df

du ·du

dx =−2 cos(3x) sin(3x)·3

Step 5: Simplify the expression:

dx =−6 cos(3x) sin(3x)

Therefore, the derivative of f(x) = cos2(3x) with respect to xis −6 cos(3x) sin(3x).

Question 25

Question

Find the derivative of the function f(x) = cos(x) sin2(x).

Solution

Step 1: Apply the product rule to find the derivative of f(x). Step 2: Let

u= cos(x) and v= sin2(x). Step 3: Find u′and v′. Step 4: Apply the

product rule formula (uv)′=u′v+uv′, where u′and v′are the derivatives of

uand vrespectively. Step 5: Substitute u,v,u′, and v′into the product rule

formula to find f′(x). Step 6: Simplify the result. Step 7: The derivative of

the function f(x) = cos(x) sin2(x) is f′(x) = −sin3(x)−2 cos(x) sin(x) cos(x),

or equivalently, f′(x) = −sin3(x)−2 cos2(x) sin(x).

Question 26

Question

Find the derivative of the following function: f(x) = sin(x)+cos(x)

sin(x)−cos(x).

Solution

Step 1: Use the quotient rule to differentiate the function. Step 2: Let u=

sin(x) + cos(x) and v= sin(x)−cos(x). Step 3: Find u′and v′. Step 4: Apply

the quotient rule to find f′(x). Step 5: Simplify the expression for f′(x).

Step 1: Use the quotient rule to differentiate the function. The quotient

rule states that if f(x) = u(x)

v(x), then f′(x) = u′(x)v(x)−u(x)v′(x)

[v(x)]2.

Step 2: Let u= sin(x) + cos(x) and v= sin(x)−cos(x).

Step 3: Find u′and v′.

u′= (sin(x) + cos(x))′= cos(x)−sin(x)

v′= (sin(x)−cos(x))′= cos(x) + sin(x)

Step 4: Apply the quotient rule to find f′(x).

f′(x) = (cos(x)−sin(x))(sin(x)−cos(x)) −(sin(x) + cos(x))(cos(x) + sin(x))

(sin(x)−cos(x))2

Step 5: Simplify the expression for f′(x).

f′(x) = (cos(x) sin(x)−cos2(x)−sin2(x) + sin(x) cos(x)) −(sin(x) cos(x) + sin2(x) + cos2(x) + cos(x) sin(x))

(sin(x)−cos(x))2

f′(x) = −2 cos2(x)−2 sin2(x)

(sin(x)−cos(x))2

f′(x) = −2(cos2(x) + sin2(x))

(sin(x)−cos(x))2

f′(x) = −2

(sin(x)−cos(x))2

Question 27

Question

Find the derivative of the function f(x) = sin(x)·cos(x).

Solution

Step 1: Apply the product rule to differentiate the function f(x) = sin(x)·cos(x).

f′(x) = (sin(x))′·cos(x) + sin(x)·(cos(x))′

Step 2: Find the derivatives of sin(x) and cos(x).

sin′(x) = cos(x) and cos′(x) = −sin(x)

Step 3: Substitute the derivatives back into f′(x).

f′(x) = (cos(x)) ·cos(x) + sin(x)·(−sin(x))

Step 4: Simplify the expression.

f′(x) = cos2(x)−sin2(x)

Step 5: Recall the trigonometric identity cos(2x) = cos2(x)−sin2(x).

f′(x) = cos(2x)

Therefore, the derivative of f(x) = sin(x)·cos(x) is f′(x) = cos(2x).

Question 28

Question

Find the derivative of the function f(x) = cos2(3x).

Solution

Step 1: Apply the chain rule by letting u= 3x, then f(x) = cos2(u).

Step 2: Find f′(u) using the chain rule, which states that d

dx [g(u)] = g′(u)·

dx .

f′(u) = 2 cos(u)·(−sin(u))

=−2 cos(u) sin(u)

Step 3: Substitute u= 3xback in to get f′(x).

f′(x) = −2 cos(3x) sin(3x)

=−sin(6x)

Therefore, the derivative of the function f(x) = cos2(3x) is f′(x) = −sin(6x).

Question 29

Question

Find the derivative of y= tan(2x) + sin(3x) with respect to x.

Solution

Step 1: Apply the derivative rules to find the derivative of tan(2x).

Step 2: Apply the chain rule, d

dx (tan(u)) = sec2(u)·du

dx , with u= 2x.

dx(tan(2x)) = sec2(2x)·2

Step 3: Simplify the derived function for tan(2x).

dx(tan(2x)) = 2 sec2(2x)

Step 4: Apply the derivative rules to find the derivative of sin(3x).

Step 5: Apply the chain rule, d

dx (sin(u)) = cos(u)·du

dx , with u= 3x.

dx(sin(3x)) = cos(3x)·3

Step 6: Simplify the derived function for sin(3x).

dx(sin(3x)) = 3 cos(3x)

Step 7: Combine the derivatives of tan(2x) and sin(3x) to find the derivative

of y.dy

dx = 2 sec2(2x) + 3 cos(3x)

Question 30

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 will use the product rule of

differentiation.

Step 1: Apply the product rule: (uv)′=u′v+uv′. Let u= sin(x) and

v= cos(x). Then, u′= cos(x) and v′=−sin(x).

Step 2: Substitute u,v,u′, and v′into the product rule formula.

f′(x) = u′v+uv′= (cos(x))(cos(x)) + (sin(x))(−sin(x))

Step 3: Simplify the expression.

f′(x) = (cos(x))2−(sin(x))2

Step 4: Recall the Pythagorean identity: cos2(x)−sin2(x) = cos(2x).

f′(x) = cos(2x)

Therefore, the derivative of the function f(x) = sin(x) cos(x) is cos(2x).

Question 31

Question

Find the derivative of f(x) = cos2(x) sin(x).

Solution

Step 1: We will use the product rule to differentiate the given function. The

product rule states that if uand vare functions of x, then the derivative of

u(x)v(x) is u(x)v′(x) + u′(x)v(x).

Step 2: Let u(x) = cos2(x) and v(x) = sin(x). Then, u′(x) = −2 cos(x) sin(x)

and v′(x) = cos(x).

Step 3: Applying the product rule, we have

f′(x) = u(x)v′(x) + u′(x)v(x)

= (cos2(x))(cos(x)) + (−2 cos(x) sin(x))(sin(x))

= cos3(x)−2 cos(x) sin2(x).

Therefore, the derivative of f(x) = cos2(x) sin(x) is f′(x) = cos3(x)−

2 cos(x) sin2(x).

Question 32

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 will use the product rule.

Step 1: Apply the product rule, which states that for two functions u(x)

and v(x), the derivative of their product is given by:

(u·v)′=u′v+uv′

Let u(x) = sin(x) and v(x) = cos(x). Then, we have:

u′(x) = cos(x) and v′(x) = −sin(x)

Step 2: Apply the product rule to find f′(x):

f′(x) = (sin(x) cos(x))′

= sin(x)(−sin(x)) + cos(x) cos(x)

=−sin2(x) + cos2(x)

= cos(2x)

Therefore, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos(2x).

Question 33

Question

Find the derivative of y= sin2(3x) + cos2(3x) with respect to x.

Solution

Step 1: Apply the chain rule for both terms.

dx(sin2(3x)) = 2 sin(3x) cos(3x)·3 = 6 sin(3x) cos(3x)

dx(cos2(3x)) = 2 cos(3x)(−sin(3x)) ·3 = −6 cos(3x) sin(3x)

Step 2: Add the derivatives of the two terms.

y′= 6 sin(3x) cos(3x)−6 cos(3x) sin(3x)

Step 3: Simplify the expression.

y′= 6 sin(3x) cos(3x)−6 cos(3x) sin(3x)

y′= 6 sin(6x)−6 sin(6x)

Step 4: Combine like terms.

y′= 0

Therefore, the derivative of y= sin2(3x) + cos2(3x) with respect to xis 0.

Question 34

Question

Calculate the derivative of the function f(x) = sin(x) cos(x).

Solution

Step 1: First, use the product rule to differentiate f(x) = sin(x) cos(x). Step

2: 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) by differentiating sin(x) and cos(x), respectively. Step 5: u′(x) = cos(x)

and v′(x) = −sin(x). Step 6: Substitute into the product rule formula to

find f′(x). Step 7: f′(x) = sin(x)(−sin(x)) + cos(x) cos(x). Step 8: Simplify

to get f′(x) = −sin2(x) + cos2(x). Step 9: Recall the Pythagorean identity:

sin2(x) + cos2(x) = 1. Step 10: Substitute the Pythagorean identity into the

expression for f′(x). Step 11: f′(x) = −1.

Therefore, the derivative of the function f(x) = sin(x) cos(x) is −1.

Question 35

Question

Find the derivative of the function f(x) = sin2(x)

xusing the quotient rule.

Question 2

Question

Find the derivative of the function f(x) = sin2(3x)−cos2(2x).

Solution

Step 1: Apply the chain rule to differentiate sin2(3x).

Let u= sin(3x) =⇒du

dx = 3 cos(3x)

Now, differentiate sin2(3x) = u2with respect to x:d

dx(sin2(3x)) = 2 sin(3x)·3 cos(3x)

= 6 sin(3x) cos(3x)

Step 2: Apply the chain rule to differentiate cos2(2x).

Let v= cos(2x) =⇒dv

dx =−2 sin(2x)

Now, differentiate cos2(2x) = v2with respect to x:d

dx(cos2(2x)) = 2 cos(2x)·−2 sin(2x)

=−4 cos(2x) sin(2x)

Step 3: Combine the derivatives of sin2(3x) and cos2(2x) to find the deriva-

tive of f(x).

f′(x) = 6 sin(3x) cos(3x)+(−4 cos(2x) sin(2x))

f′(x) = 2 sin(3x) cos(3x)−4 cos(2x) sin(2x)

Therefore, the derivative of f(x) = sin2(3x)−cos2(2x) is f′(x) = 2 sin(3x) cos(3x)−

4 cos(2x) sin(2x).

Question 3

Question

Find the derivative of the function f(x) = cos2(3x).

Solution

Step 1: Apply the chain rule, which states that if uand vare both differentiable

functions of x, then the derivative of u(v(x)) with respect to xis u′(v(x))·v′(x).

Step 2: Let u= cos(x) and v= 3x. Then f(x) = u2(v) = u(v(x))2=

(cos(3x))2.

Step 3: Find f′(x) using the chain rule and the derivative of cos(x).

f′(x) = 2 cos(3x)·(cos(3x))′

= 2 cos(3x)·(−3 sin(3x))

=−6 cos(3x) sin(3x)

Therefore, the derivative of f(x) = cos2(3x) is f′(x) = −6 cos(3x) sin(3x).

Question 4

Question

Find the derivative of the function f(x) = sin(2x) cos(x).

Solution

Step 1: Apply the product rule to find the derivative of f(x). Step 2: Recall

that the product rule states that the derivative of the product of two functions

is given by (fg)′=f′g+fg′. Step 3: Let f(x) = sin(2x) and g(x) = cos(x).

Step 4: Find f′(x) and g′(x). Step 5: The derivative of sin(2x) with respect to

xis d

dx (sin(2x)) = 2 cos(2x) by the chain rule. Step 6: The derivative of cos(x)

with respect to xis d

dx (cos(x)) = −sin(x). Step 7: Now, apply the product rule

to find f′(x) and g′(x). Step 8: f′(x) = cos(2x)·2 by the chain rule. Step 9:

g′(x) = −sin(x). Step 10: Now, put all the pieces together using the product

rule: f′(x)g(x) + f(x)g′(x). Step 11: So, the derivative of f(x) = sin(2x) cos(x)

is 2 cos(2x) cos(x)−sin(2x) sin(x).

Question 5

Question

Find the derivative of the function f(x) = sin3(2x) + cos2(x).

Solution

Step 1: To find the derivative of f(x), we will differentiate each term separately

using the rules of differentiation.

Step 2: Let’s start by finding the derivative of sin3(2x). Using the chain

rule, the derivative of sin3(2x) will be:

dx sin3(2x) = 3 sin2(2x) cos(2x)·2 = 6 sin2(2x) cos(2x).

Step 3: Next, let’s find the derivative of cos2(x). The derivative of cos2(x)

will be: d

dx cos2(x) = 2 cos(x)(−sin(x)) = −2 cos(x) sin(x).

Step 4: Therefore, the derivative of f(x) = sin3(2x) + cos2(x) will be:

f′(x) = 6 sin2(2x) cos(2x)−2 cos(x) sin(x).

So, the derivative of f(x) is f′(x) = 6 sin2(2x) cos(2x)−2 cos(x) sin(x).

Question 6

Question

Compute the derivative of f(x) = sin2(3x) + cos2(4x).

Solution

Step 1: Use the trigonometric identity sin2(θ) + cos2(θ) = 1 to simplify f(x).

f(x) = sin2(3x) + cos2(4x)

= 1

Step 2: Since f(x) = 1 for all x, the derivative of f(x) is 0.

f′(x)=0

Question 7

Question

Find the derivative of the function f(x) = cos(3x) sin(4x).

Solution

Step 1: Apply the product rule to differentiate the function. Step 2: Use the

chain rule to differentiate the trigonometric functions. Step 3: Simplify the

expression to get the final derivative.

Step 1:

Apply the product rule, which states that (uv)′=u′v+uv′, where uand vare

functions of x. Let u(x) = cos(3x) and v(x) = sin(4x). Then, the derivative of

f(x) is given by:

f′(x) = u′(x)v(x) + u(x)v′(x)

Step 2:

Now, differentiate u(x) = cos(3x) and v(x) = sin(4x). We have:

u′(x) = −3 sin(3x)

v′(x) = 4 cos(4x)

Step 3:

Now, substitute these derivatives back into the product rule formula:

f′(x)=(−3 sin(3x)) sin(4x) + cos(3x)(4 cos(4x))

f′(x) = −3 sin(3x) sin(4x) + 4 cos(3x) cos(4x)

f′(x) = −sin(7x) + cos(x)

Therefore, the derivative of the function f(x) = cos(3x) sin(4x) is f′(x) =

−sin(7x) + 4 cos(3x) cos(4x).

Question 8

Question

Find the derivative of f(x) = sinx2cos(2x).

Solution

Step 1: We will use the product rule to differentiate the given function. 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x2and h(x) = cos(2x).

Step 3: Calculate g′(x) and h′(x).

g′(x) = (sinx2)′

= cosx2·(x2)′

= cosx2·2x

= 2xcosx2

Step 4:

h′(x) = (cos(2x))′

=−sin(2x)·(2x)′

=−2 sin(2x)

Step 5: Apply the product rule:

f′(x) = (2xcosx2)(cos(2x)) + (sinx2)(−2 sin(2x))

= 2xcosx2cos(2x)−2 sinx2sin(2x)

Therefore, the derivative of f(x) = sinx2cos(2x) is f′(x) = 2xcosx2cos(2x)−

2 sinx2sin(2x).

Question 9

Question

Find the derivative of f(x) = sin(x) cos(x).

Solution

Step 1: Use the product rule to differentiate f(x) = sin(x) cos(x). Step 2: Let

u= sin(x) and v= cos(x). Step 3: Find u′and v′. Step 4: Apply the product

rule: f′(x) = u′v+uv′. Step 5: Substitute the values of u,v,u′, and v′back

into the formula to find the derivative of f(x).

Let’s start solving the problem:

1. Step 1: Apply the product rule:

f′(x) = (sin(x))′(cos(x)) + sin(x)(cos(x))′.

2. Step 2: Let u= sin(x) and v= cos(x).

3. Step 3: Find u′and v′:

u′= cos(x) and v′=−sin(x).

4. Step 4: Apply the product rule:

f′(x) = (cos(x) cos(x)) + (sin(x)(−sin(x))).

5. Step 5: Calculate the derivatives and simplify:

f′(x) = cos2(x)−sin2(x).

Therefore, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos2(x)−

sin2(x).

Question 10

Question

Find the derivative of y= sin2(3x) + cos2(2x) with respect to x.

Solution

Step 1: Apply the chain rule for sin2(3x).

Step 2: Apply the chain rule for cos2(2x).

Step 3: Combine the derivatives of both terms to find the overall derivative

of y.

Step 1: We have y= sin2(3x), so let u= 3x. Thus, we can rewrite yas

y= (sin u)2. Applying the chain rule gives:

dx = 2 sin(3x) cos(3x)·3 = 6 sin(3x) cos(3x)

Step 2: We have y= cos2(2x), so let v= 2x. Thus, we can rewrite yas

y= (cos v)2. Applying the chain rule gives:

dx = 2 cos(2x) sin(2x)·2 = 4 cos(2x) sin(2x)

Step 3: Combining the derivatives of both terms gives:

dx = 6 sin(3x) cos(3x) + 4 cos(2x) sin(2x) = 6 sin(6x) + 4 sin(4x)

Question 11

Question

Calculate the derivative of the function f(x) = sin(x) cos(x).

Solution

To find the derivative of the function f(x) = sin(x) cos(x), we will use the

product rule.

Step 1: Apply the product rule. Let u= sin(x) and v= cos(x). Then,

u′= cos(x) and v′=−sin(x).

Step 2: Use the product rule formula: (uv)′=u′v+uv′.

f′(x) = (sin(x) cos(x))′

= sin(x)·(−sin(x)) + cos(x)·cos(x)

=−sin2(x) + cos2(x)

Step 3: Use the Pythagorean identity sin2(x) + cos2(x) = 1.

f′(x) = −(1) = −1

Therefore, the derivative of f(x) = sin(x) cos(x) is −1.

Question 12

Question

Determine the derivative of f(x) = cos2(x) tan(x).

Solution

To find the derivative of f(x), we will use the product rule and the chain rule

for differentiation.

f(x) = cos2(x) tan(x)

= (cos(x) cos(x))(sin(x)/cos(x))

= cos(x) cos(x)·sin(x)

cos(x)

= cos(x)·cos(x)·sin(x)

cos(x)

= cos(x)·sin(x)

= sin(x) cos(x)

Now, we can differentiate f(x):

f′(x) = (sin(x))′·(cos(x)) + (cos(x))′·(sin(x))

= cos(x) cos(x)+(−sin(x)) sin(x)

= cos2(x)−sin2(x)

Therefore, the derivative of f(x) = cos2(x) tan(x) is f′(x) = cos2(x)−

sin2(x).

Question 13

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 f(x). Step 2: Use the

chain rule to differentiate sin(2x) and cos(3x). Step 3: Simplify the expression

and combine like terms to get the final answer.

Step 1:

Apply the product rule:

f′(x) = d

dx (sin(2x) cos(3x)) = sin(2x)d

dx cos(3x) + cos(3x)d

dx sin(2x)

Step 2:

Differentiate sin(2x) and cos(3x) using the chain rule:

dx sin(2x) = cos(2x)·2

dx cos(3x) = −sin(3x)·3

Step 3:

Substitute the derivatives back into the expression:

f′(x) = sin(2x)(−3 sin(3x)) + cos(3x)(2 cos(2x))

f′(x) = −3 sin(2x) sin(3x) + 2 cos(3x) cos(2x)

Therefore, the derivative of f(x) = sin(2x) cos(3x) is f′(x) = −3 sin(2x) sin(3x)+

2 cos(3x) cos(2x).

Question 14

Question

Find the derivative of the function f(x) = sin2(2x) + cos2(3x).

Solution

Step 1: Apply the chain rule to differentiate sin2(2x).

dx(sin2(2x)) = 2 sin(2x) cos(2x)

Step 2: Apply the chain rule to differentiate cos2(3x).

dx(cos2(3x)) = −2 cos(3x) sin(3x)

Step 3: Combine the derivatives of each term to find the derivative of the

whole function f(x).

f′(x) = 2 sin(2x) cos(2x)−2 cos(3x) sin(3x)

Therefore, the derivative of f(x) = sin2(2x)+cos2(3x) is f′(x) = 2 sin(2x) cos(2x)−2 cos(3x) sin(3x) .

Question 15

Question

Find the derivative of the function f(x) = tan(x)−sin(x).

Solution

Step 1: Recall the derivatives of the trigonometric functions:



dx (sin(x)) = cos(x)



dx (cos(x)) = −sin(x)



dx (tan(x)) = sec2(x)

Step 2: Use the derivative rules to find f′(x):

f′(x) = d

dx(tan(x)) −d

dx(sin(x))

Step 3: Substitute the derivatives of tan(x) and sin(x):

f′(x) = sec2(x)−cos(x)

So, the derivative of f(x) = tan(x)−sin(x) is f′(x) = sec2(x)−cos(x).

Question 16

Question

Find the derivative of the function f(x) = sin2(3x)−cos2(2x).

Solution

To find the derivative f′(x) of the given function f(x), we will use the chain

rule and the power rule for differentiation.

Step 1: Find the derivative of sin2(3x). Let u= sin(3x). Then, u2=

sin2(3x). Using the chain rule, we have:

dx[sin2(3x)] = d

du(u2)·du

dx = 2ucos(3x)·3.

Thus, the derivative of sin2(3x) is 2 sin(3x) cos(3x)·3.

Step 2: Find the derivative of cos2(2x). Let v= cos(2x). Then, v2=

cos2(2x). Using the chain rule, we have:

dx[cos2(2x)] = d

dv (v2)·dv

dx = 2v(−sin(2x)) ·2.

Thus, the derivative of cos2(2x) is −4 cos(2x) sin(2x).

Step 3: Combine the derivatives. The derivative of the given function

f(x) = sin2(3x)−cos2(2x) is:

f′(x) = 2 sin(3x) cos(3x)·3+(−4 cos(2x) sin(2x)).

So, the derivative of f(x) is 6 sin(3x) cos(3x)−4 cos(2x) sin(2x).

Question 17

Question

Find the derivative of y=sin(2x)

cos(x).

Solution

To find the derivative of y=sin(2x)

cos(x), we will use the quotient rule and the chain

rule of differentiation.

Step 1: Apply the quotient rule, which states that if uand vare differen-

tiable functions, then u

v′=u′v−uv′

v2. Let u= sin(2x) and v= cos(x). Then,

u′= 2 cos(2x) and v′=−sin(x).

Step 2: Compute y′using the quotient rule.

y′=(2 cos(2x)·cos(x)−sin(2x)· − sin(x))

cos2(x)

=2 cos(2x) cos(x) + sin(2x) sin(x)

cos2(x)

Step 3: Apply the angle sum identities cos(a+b) = cos(a) cos(b)−sin(a) sin(b)

and sin(a+b) = sin(a) cos(b) + cos(a) sin(b) to simplify the expression.

y′=2(cos(x) cos(2x) + sin(x) sin(2x))

cos2(x)

Step 4: Use the angle sum identities again to further simplify.

y′=2 cos(x+ 2x)

cos2(x)=2 cos(3x)

cos2(x)

Therefore, the derivative of y=sin(2x)

cos(x)is y′=2 cos(3x)

cos2(x).

Question 18

Question

Find the derivative of the function f(x) = cos(2x) sin(3x).

Solution

Step 1: Apply the product rule to find the derivative of f(x). Step 2: Recall

that the product rule states (u·v)′=u′v+uv′. In this case, let u= cos(2x) and

v= sin(3x). Step 3: Find u′and v′. Step 4: u′=−2 sin(2x) and v′= 3 cos(3x).

Step 5: Now apply the product rule to find f′(x). Step 6: f′(x) = cos(2x)·

3 cos(3x)+ (−2 sin(2x))·sin(3x). Step 7: Simplify the expression to get the final

answer. Step 8: f′(x) = 3 cos(2x) cos(3x)−2 sin(2x) sin(3x).

Question 19

Question

Find the derivative of the function f(x) = sin(2x) cos(3x).

Solution

To find the derivative of f(x) = sin(2x) cos(3x), we will use the product rule.

Step 1: Apply the product rule. If f(x) = g(x)·h(x), then f′(x) = g′(x)·

h(x) + g(x)·h′(x).

Let g(x) = sin(2x) and h(x) = cos(3x).

Step 2: Find g′(x) and h′(x).



Calculate g′(x) by applying the chain rule: g′(x) = 2 cos(2x).



Calculate h′(x) by applying the chain rule: h′(x) = −3 sin(3x).

Step 3: Substitute g(x), g′(x), h(x), and h′(x) back into the product rule

formula.

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 f(x) = sin(2x) cos(3x) is f′(x) = 2 cos(2x) cos(3x)−

3 sin(2x) sin(3x).

Question 20

Question

Find the derivative of y=cos(x) sin(x)

Solution

Step 1: To find the derivative using the quotient rule, let u(x) = cos(x) sin(x)

and v(x) = x.

Step 2: Find u′(x).

u′(x) = (cos(x))′(sin(x)) + cos(x)(sin(x))′= (−sin(x)) sin(x) + cos(x) cos(x)

u′(x) = −sin2(x) + cos2(x)

Step 3: Find v′(x).

v′(x)=1

Step 4: Apply the quotient rule.

dx u(x)

v(x)=u′(x)v(x)−u(x)v′(x)

(v(x))2

dx cos(x) sin(x)

x=(−sin2(x) + cos2(x))x−(cos(x) sin(x))(1)

Step 5: Simplify the expression.

dx cos(x) sin(x)

x=−xsin2(x) + xcos2(x)−cos(x) sin(x)

dx cos(x) sin(x)

x=x(cos2(x)−sin2(x)) −cos(x) sin(x)

Hence, the derivative of y=cos(x) sin(x)

xis y′=x(cos2(x)−sin2(x))−cos(x) sin(x)

x2.

Question 21

Question

Find the derivative of y= sin(2x) cos(3x).

Solution

Step 1: Apply the product rule to differentiate the given function.

Step 2: Let u= sin(2x) and v= cos(3x).

Step 3: Find u′and v′.

u′=d

dx(sin(2x)) = 2 cos(2x)

v′=d

dx(cos(3x)) = −3 sin(3x)

Step 4: Apply the product rule to get y′.

y′=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) is 2 cos(2x) cos(3x)−3 sin(2x) sin(3x).

Question 22

Question

Find the derivative of the function f(x) = sin(3x) cos(2x).

Solution

Step 1: Apply the product rule. Step 2: Recall 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(3x)

and v(x) = cos(2x). Step 4: Find u′(x) and v′(x). Step 5: u′(x) = 3 cos(3x) by

applying the chain rule. Step 6: v′(x) = −2 sin(2x) by applying the chain rule.

Step 7: Substitute u(x), v(x), u′(x), and v′(x) into the product rule formula.

Step 8: f′(x) = (3 cos(3x) cos(2x))+(sin(3x)(−2 sin(2x))). Step 9: Simplify the

expression. Step 10: f′(x) = 3 cos(3x) cos(2x)−2 sin(3x) sin(2x).

Question 23

Question

Find the derivative of the function f(x) = cos2(x) + sin(2x).

Solution

Step 1: We will differentiate each term in the function f(x) separately using the

rules of differentiation.

Step 2: For the first term, cos2(x), we will use the chain rule. Let u= cos(x).

Then, f(x) = u2and df

dx = 2udu

dx .

Step 3: Differentiating cos(x) with respect to xgives d

dx (cos(x)) = −sin(x).

Step 4: Applying the chain rule, the derivative of cos2(x) is d

dx (cos2(x)) =

2 cos(x)(−sin(x)).

Step 5: Simplifying, we have d

dx (cos2(x)) = −2 cos(x) sin(x).

Step 6: For the second term sin(2x), we can use the trigonometric identity

sin(2x) = 2 sin(x) cos(x).

Step 7: So, the derivative of sin(2x) is d

dx (sin(2x)) = d

dx (2 sin(x) cos(x)) =

2(cos(x) cos(x)−sin(x) sin(x)).

Step 8: Simplifying, we get d

dx (sin(2x)) = 2(cos2(x)−sin2(x)).

Step 9: Therefore, the derivative of the function f(x) = cos2(x) + sin(2x) is:

dx(f(x)) = −2 cos(x) sin(x) + 2(cos2(x)−sin2(x))

Question 24

Question

Differentiate the function f(x) = cos2(3x) with respect to x.

Solution

Step 1: Apply the chain rule by letting u= 3x. Step 2: Calculate df

du . Step

3: Calculate du

dx . Step 4: Substitute the results from Step 2 and Step 3 into

the chain rule formula. Step 5: Simplify the expression to find the derivative of

f(x).

Step 1: Let u= 3x, so f(x) = cos2(u).

Step 2: Calculate df

du by differentiating cos2(u) with respect to u:

du = 2 cos(u)·(−sin(u)) = −2 cos(u) sin(u)

Step 3: Calculate du

dx :

dx = 3

Step 4: Apply the chain rule:

dx =df

du ·du

dx =−2 cos(3x) sin(3x)·3

Step 5: Simplify the expression:

dx =−6 cos(3x) sin(3x)

Therefore, the derivative of f(x) = cos2(3x) with respect to xis −6 cos(3x) sin(3x).

Question 25

Question

Find the derivative of the function f(x) = cos(x) sin2(x).

Solution

Step 1: Apply the product rule to find the derivative of f(x). Step 2: Let

u= cos(x) and v= sin2(x). Step 3: Find u′and v′. Step 4: Apply the

product rule formula (uv)′=u′v+uv′, where u′and v′are the derivatives of

uand vrespectively. Step 5: Substitute u,v,u′, and v′into the product rule

formula to find f′(x). Step 6: Simplify the result. Step 7: The derivative of

the function f(x) = cos(x) sin2(x) is f′(x) = −sin3(x)−2 cos(x) sin(x) cos(x),

or equivalently, f′(x) = −sin3(x)−2 cos2(x) sin(x).

Question 26

Question

Find the derivative of the following function: f(x) = sin(x)+cos(x)

sin(x)−cos(x).

Solution

Step 1: Use the quotient rule to differentiate the function. Step 2: Let u=

sin(x) + cos(x) and v= sin(x)−cos(x). Step 3: Find u′and v′. Step 4: Apply

the quotient rule to find f′(x). Step 5: Simplify the expression for f′(x).

Step 1: Use the quotient rule to differentiate the function. The quotient

rule states that if f(x) = u(x)

v(x), then f′(x) = u′(x)v(x)−u(x)v′(x)

[v(x)]2.

Step 2: Let u= sin(x) + cos(x) and v= sin(x)−cos(x).

Step 3: Find u′and v′.

u′= (sin(x) + cos(x))′= cos(x)−sin(x)

v′= (sin(x)−cos(x))′= cos(x) + sin(x)

Step 4: Apply the quotient rule to find f′(x).

f′(x) = (cos(x)−sin(x))(sin(x)−cos(x)) −(sin(x) + cos(x))(cos(x) + sin(x))

(sin(x)−cos(x))2

Step 5: Simplify the expression for f′(x).

f′(x) = (cos(x) sin(x)−cos2(x)−sin2(x) + sin(x) cos(x)) −(sin(x) cos(x) + sin2(x) + cos2(x) + cos(x) sin(x))

(sin(x)−cos(x))2

f′(x) = −2 cos2(x)−2 sin2(x)

(sin(x)−cos(x))2

f′(x) = −2(cos2(x) + sin2(x))

(sin(x)−cos(x))2

f′(x) = −2

(sin(x)−cos(x))2

Question 27

Question

Find the derivative of the function f(x) = sin(x)·cos(x).

Solution

Step 1: Apply the product rule to differentiate the function f(x) = sin(x)·cos(x).

f′(x) = (sin(x))′·cos(x) + sin(x)·(cos(x))′

Step 2: Find the derivatives of sin(x) and cos(x).

sin′(x) = cos(x) and cos′(x) = −sin(x)

Step 3: Substitute the derivatives back into f′(x).

f′(x) = (cos(x)) ·cos(x) + sin(x)·(−sin(x))

Step 4: Simplify the expression.

f′(x) = cos2(x)−sin2(x)

Step 5: Recall the trigonometric identity cos(2x) = cos2(x)−sin2(x).

f′(x) = cos(2x)

Therefore, the derivative of f(x) = sin(x)·cos(x) is f′(x) = cos(2x).

Question 28

Question

Find the derivative of the function f(x) = cos2(3x).

Solution

Step 1: Apply the chain rule by letting u= 3x, then f(x) = cos2(u).

Step 2: Find f′(u) using the chain rule, which states that d

dx [g(u)] = g′(u)·

dx .

f′(u) = 2 cos(u)·(−sin(u))

=−2 cos(u) sin(u)

Step 3: Substitute u= 3xback in to get f′(x).

f′(x) = −2 cos(3x) sin(3x)

=−sin(6x)

Therefore, the derivative of the function f(x) = cos2(3x) is f′(x) = −sin(6x).

Question 29

Question

Find the derivative of y= tan(2x) + sin(3x) with respect to x.

Solution

Step 1: Apply the derivative rules to find the derivative of tan(2x).

Step 2: Apply the chain rule, d

dx (tan(u)) = sec2(u)·du

dx , with u= 2x.

dx(tan(2x)) = sec2(2x)·2

Step 3: Simplify the derived function for tan(2x).

dx(tan(2x)) = 2 sec2(2x)

Step 4: Apply the derivative rules to find the derivative of sin(3x).

Step 5: Apply the chain rule, d

dx (sin(u)) = cos(u)·du

dx , with u= 3x.

dx(sin(3x)) = cos(3x)·3

Step 6: Simplify the derived function for sin(3x).

dx(sin(3x)) = 3 cos(3x)

Step 7: Combine the derivatives of tan(2x) and sin(3x) to find the derivative

of y.dy

dx = 2 sec2(2x) + 3 cos(3x)

Question 30

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 will use the product rule of

differentiation.

Step 1: Apply the product rule: (uv)′=u′v+uv′. Let u= sin(x) and

v= cos(x). Then, u′= cos(x) and v′=−sin(x).

Step 2: Substitute u,v,u′, and v′into the product rule formula.

f′(x) = u′v+uv′= (cos(x))(cos(x)) + (sin(x))(−sin(x))

Step 3: Simplify the expression.

f′(x) = (cos(x))2−(sin(x))2

Step 4: Recall the Pythagorean identity: cos2(x)−sin2(x) = cos(2x).

f′(x) = cos(2x)

Therefore, the derivative of the function f(x) = sin(x) cos(x) is cos(2x).

Question 31

Question

Find the derivative of f(x) = cos2(x) sin(x).

Solution

Step 1: We will use the product rule to differentiate the given function. The

product rule states that if uand vare functions of x, then the derivative of

u(x)v(x) is u(x)v′(x) + u′(x)v(x).

Step 2: Let u(x) = cos2(x) and v(x) = sin(x). Then, u′(x) = −2 cos(x) sin(x)

and v′(x) = cos(x).

Step 3: Applying the product rule, we have

f′(x) = u(x)v′(x) + u′(x)v(x)

= (cos2(x))(cos(x)) + (−2 cos(x) sin(x))(sin(x))

= cos3(x)−2 cos(x) sin2(x).

Therefore, the derivative of f(x) = cos2(x) sin(x) is f′(x) = cos3(x)−

2 cos(x) sin2(x).

Question 32

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 will use the product rule.

Step 1: Apply the product rule, which states that for two functions u(x)

and v(x), the derivative of their product is given by:

(u·v)′=u′v+uv′

Let u(x) = sin(x) and v(x) = cos(x). Then, we have:

u′(x) = cos(x) and v′(x) = −sin(x)

Step 2: Apply the product rule to find f′(x):

f′(x) = (sin(x) cos(x))′

= sin(x)(−sin(x)) + cos(x) cos(x)

=−sin2(x) + cos2(x)

= cos(2x)

Therefore, the derivative of f(x) = sin(x) cos(x) is f′(x) = cos(2x).

Question 33

Question

Find the derivative of y= sin2(3x) + cos2(3x) with respect to x.