topic badge

6.07 Differentiating composite functions

Worksheet
Composite functions
1

Differentiate the following:

a
y=e^x\sin x
b
y = e^{\sin x}
c
y = e^{x} \cos 3x
d
y = e^{ 3 x} \cos \left(\dfrac{x}{3}\right)
e
y = e^{ 2 x} \tan 5 x
f
y = e^{ - x } \sin 4 x
g
y = \cos \left(\ln x\right)
h
y = \cos \left(\ln \left( - 5 x \right)\right)
i
y = \cos x \ln x
j
y = \tan x \ln \left( 4 x\right)
k
y = e^{x} \ln x
l
y = e^{ 2 x} \ln \left( 4 x\right)
m
y = \log_{e} \left(\ln x\right)
n
y = \dfrac{\ln x}{e^{ 2 x}}
o
y = e^{ - 2 } \ln \left( - 6 + x^{ - 3 }\right)
p
y = \cos \left(\dfrac{\pi t}{2} - \dfrac{\pi}{3}\right)
q
y = x^{2} \sin \left(\dfrac{1}{x}\right)
r
y = 4 \sin \left(\dfrac{x}{5}\right) - 6 e^{ 2 x} + x^{ - 8 }
s
y = \ln \left(x^{2} - 7 x - 12\right) - \sqrt{ 11 x}
t
y = \left(e^{ - 5 x^{2} } + \cos x\right)^{5}
u
y = \dfrac{e^{ - 0.2 x}}{\sin \left( \dfrac{\pi}{4} x\right) - x^{4}}
v
y = \left(\cos x + \sin x\right) e^{ 6 x}
w
y = \dfrac{\cos \left( 4 x - \dfrac{10 \pi}{11}\right)}{\left(x + 3\right)^{2}}
2

Find the derivative of the following:.

a
e^{ 5 x}\cos \left(x\right)
b
x^{3} \sin x
c
\cos ^{2}\left(x\right)
d
\dfrac{\cos x - \sin x}{\cos x + \sin x}
e
\dfrac{4 x}{\sin x}
f
e^{ 3 x} \cos \left( 5 x + \dfrac{4 \pi}{7}\right)
g
\dfrac{\ln x}{\sin x}
h
\ln \left(\sin x\right)
3

For each of the following curves and given points:

i

Find an expression for \dfrac{dy}{dx}.

ii

Find the exact value of the gradient of the curve at the given point.

a

y = \cos 3 x at x = \dfrac{\pi}{18}.

b
y = 6- \cos 3 x at x = \dfrac{\pi}{9}.
c

y = \sin 4 x at x = \dfrac{\pi}{16}.

d

y = \sin ^{2}\left( 4 x\right) at x = \dfrac{\pi}{32}.

e

y = \cos ^{2}\left( 2 x\right) at x = \dfrac{\pi}{24}.

4

Consider the equation y = e^{\tan \left( - 6 x\right)}.

a

Let f \left( x \right) = \tan \left( - 6 x\right). Find f' \left( x \right).

b

Hence, differentiate y = e^{\tan \left( - 6 x\right)}.

5

Consider the expression e^{\cos x} \sin \left(e^{x}\right).

a

If u = e^{\cos x}, find \dfrac{d u}{d x}.

b

If v = \sin \left(e^{x}\right), find \dfrac{d v}{d x}.

c

Hence, find the derivative of y = e^{\cos x} \sin \left(e^{x}\right).

6

Consider the function y = \dfrac{4 x^{2} + e^{x}}{\cos 7 x}.

a

If u = 4 x^{2} + e^{x}, find u'.

b

If v = \cos 7x, find v'.

c

Hence, find y'.

7

Consider the function y = x e^{x}.

a

Show that e^{x + \ln x} = x e^{x}.

b

Hence, find \dfrac{d y}{d x}, without using the product rule.

8

Determine \dfrac{d y}{d x}, given that y = u^{5} and u = \ln \left(x + 5\right).

9

Find the derivative of the following:

a

y = \dfrac{x}{3} e^{ - 3 x}

b

y = \left( 4 x + 6\right) \tan 4 x

c

g \left( x \right) = \dfrac{x}{3} e^{ - 3 x} - \left( 4 x + 6\right) \tan 4 x

10

Consider g \left( x \right) = \sqrt { \left( e^{ 3 x} + x^{ - 4 } - \tan \dfrac{\pi}{4} x \right) } .

a

Find the derivative of y = e^{ 3 x} + x^{ - 4 } - \tan \dfrac{\pi}{4} x

b

Hence, differentiate g \left( x \right) = \sqrt{y}, expressing your answer in exact form.

11

Consider the expression e^{x} \left(1 + \ln 0.3 x\right)^{ - 2 }.

a

If y = \left(1 + \ln \left( 0.3 x\right)\right)^{ - 2 }, find \dfrac{d y}{d x}.

b

Hence, find the derivative of y = e^{x} \left(1 + \ln 0.3 x\right)^{ - 2 }.

12

Consider the equation y = \left( 2 \ln \left( - 3 x - x^{2}\right) - \cos \left( \dfrac{\pi}{3} x\right) + \dfrac{1}{x^{4}}\right)^{3}.

a

Find the derivative of 2 \ln \left( - 3 x - x^{2}\right).

b

Hence differentiate y = \left( 2 \ln \left( - 3 x - x^{2}\right) - \cos \left( \dfrac{\pi}{3} x\right) + \dfrac{1}{x^{4}}\right)^{3}.

13

Consider the function y = \dfrac{\sin ^{2}\left(x\right)}{\cos x}.

a

Prove that \dfrac{\sin ^{2}\left(x\right)}{\cos x} = \sec x - \cos x.

b

Differentiate f \left( x \right) = \sec x.

c

Hence find \dfrac{dy}{dx}, without using the product or quotient rule.

Stationary points
14

The graph of the function f \left( x \right) = e^{ 2 x} \sin 3 x is shown:

a

Find f' \left( x \right).

b

Find the x-intercepts, B and C.

c

Determine the coordinates of point A, correct to two decimal places.

0.2
0.4
0.6
0.8
1
x
1
2
3
4
5
f(x)
15

Consider the function f \left( x \right) = x^{3} \ln x, over the domain e^{ - \frac{1}{2} } \leq x \leq e^{ - \frac{1}{4} }.

a

Find an expression for f' \left( x \right).

b

Find the exact values of x such that f' \left( x \right)=0.

c

Complete the table of values:

xe^{ - \frac{1}{2} }e^{ - \frac{1}{3} }e^{ - \frac{1}{4} }
f (x)
d

Hence, state the nature and location of any stationary points.

e

Calculate the exact global minimum over the domain.

f

Calculate the exact global maximum over the domain.

16

Consider the function y = \dfrac{\ln 3 x}{e^{ 3 x}}.

a

Find the x-intercept of the function.

b

Find an expression for \dfrac{d y}{d x}.

c

Show that the turning point of the graph occurs when 3 x \ln 3 x - 1 = 0.

d

An approximation to the solution of the equation 3 x \ln 3 x - 1 = 0 is x = 0.58.

Complete the table of values, correct to two decimal places.

x0.010.581.58
\dfrac{dy}{dx} 0
e

Hence, state the nature of the turning point at x = 0.58.

f

State the coordinates of the turning point, correct to two decimal places.

17

Determine the values of the non-zero constants, a and b, for the following function, given it has a turning point at \left(0.25, 1\right):

f \left( x \right) = a x e^{ b x}
18

Consider the function f \left( x \right) = \ln \left(\sin 2 x\right).

a

State the values of x between 0 and 2 \pi for which the function is defined.

b

Determine the values of x in the domain 0 \leq x \leq 2 \pi, for which the function has a maximum value.

c

State the maximum value of the function.

Applications
19

The curve y = x \cos x passes through the point Q, \left(\dfrac{\pi}{2}, 0\right).

Find the equation of the tangent at point Q.

20

Find the equation of the tangent to the following curves:

a

y = e^{x} - 3 \sin x at x = \dfrac{3 \pi}{2}.

b

y = e^{\cos x} at x = \dfrac{3 \pi}{2}.

21

Researchers have created a model to project the country’s population for the next 10 years, where P is the population (in thousands), t years from now. The model is defined by the function: P \left( t \right) = \dfrac{57\,460 e^{\frac{t}{7}}}{t + 13}

a

State the current population of the country.

b

According to the model, state the current rate of growth of the population, to the nearest thousand.

c

Find the rate of population growth 7 months from now, to the nearest thousand.

d

Find the rate of population growth 10 years from now, to the nearest thousand.

22

The displacement of a particle moving in rectilinear motion is given by: x \left( t \right) = \left(t - 2\right)^{2} + \sin \left(t - 4 \pi\right) + 3 t + 25

a

State the initial displacement of the particle.

b

Write an expression for the velocity of the particle, v \left( t \right) = x' \left( t \right) .

c

Write an expression for the acceleration of the particle, a \left( t \right) = v' \left( t \right).

23

A charged particle moves back and forth about the fixed point x = 0 (called the origin). Its position, x \text{ cm} from the origin, after t seconds is given by the equation:

x = \sin \left( \pi e^{ 2 t}\right)
a

Find the particle's initial position.

b

Find an expression for the velocity of the particle , v \left( t \right) = x' \left( t \right).

c

Find the exact times for the first and second occasion that the particle comes to a stop.

d

Describe the position of the particle when it first comes to a stop, v \left( t \right) = 0.

e

Describe the position of the particle when it comes to a stop for the second time.

Sign up to access Worksheet
Get full access to our content with a Mathspace account

Outcomes

MA12-3

applies calculus techniques to model and solve problems

MA12-6

applies appropriate differentiation methods to solve problems

What is Mathspace

About Mathspace