Topics
5. Differentiation
topic badge
AustraliaVIC
VCE 12 Methods 2023

5.07 Further differentiation

Lesson
Worksheet
Practice
Worksheet
Further differentiation
1

Differentiate each of the following:

a
y = \left(x + 7\right)^{5} - \dfrac{3}{x^{4}}
b
y = x^{4} - 4 x + 6 \sqrt{x - 7}
c
y = x^{7} + 3 x^{4} + \sqrt[3]{x + 4}
d
y = \dfrac{3}{x + 7} + \dfrac{6 x + 7}{3}
e
y = 5x^6+\dfrac{1}{x-8}+3
f
y = 5x^7\left(x^2+7\right)^3
g
y=\sqrt{\dfrac{3+7x}{3-7x}}
h
y=\dfrac{\left(x+2\right)^8}{\left(x+8\right)^3}
2

Differentiate the following, expressing your answer in factorised form where possible:

a
y = \dfrac{e^{ 3 x} + 5}{e^{ 3 x} - 5}
b
y = e^{ 3 x - 4} \left(x + 2\right)^{2}
c
y = e^{ - 4 x } \sqrt{x + 2}
d
y = x^{3} + x^{2} e^{ 3 x}
3

Differentiate the following:

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

Find the derivative of y = \tan \left(\dfrac{x + 2}{x - 2}\right) using the substitution u = \dfrac{x + 2}{x - 2}.

5

Differentiate the following:

a
y=e^x\sin x
b
y = e^{\sin x}
c
y = e^{x} \cos 3x
d
y = e^{ 5 x}\cos \left(x\right)
e
y = e^{ 3 x} \cos \left(\dfrac{x}{3}\right)
f
y=e^{ 3 x} \cos \left( 5 x + \dfrac{4 \pi}{7}\right)
g
y = e^{ 2 x} \tan 5 x
h
y = e^{ - x } \sin 4 x
i
y = \dfrac{e^{ x }} {\cos x}
j
y = 4 \sin \left(\dfrac{x}{5}\right) - 6 e^{ 2 x} + x^{ - 8 }
k
y = \left(e^{ - 5 x^{2} } + \cos x\right)^{5}
l
y = \dfrac{e^{ - 0.2 x}}{\sin \left( \dfrac{\pi}{4} x\right) - x^{4}}
m
y = \left(\cos x + \sin x\right) e^{ 6 x}
n

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

6

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)}.

7

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).

8

Consider the equation 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'.

9

Consider the function 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.

Gradients and tangents
10

Find the gradient of f \left( x \right) = \left(x - 8\right)^{3} \left(x - 4\right)^{4} at x = 6.

11

Find the gradient of f \left( x \right) = \dfrac{\left(x + 7\right)^{9}}{\left(x + 9\right)^{4}} at x = - 5.

12

Consider the function h \left( t \right) = \left(3 + t\right) \left(3 - t^{3}\right)^{4}.

a

Determine the expression for h' \left( t \right).

b

Hence, calculate the gradient of the function h \left( t \right) where t = 1.

13

Consider the function h \left( t \right) = \left(1 - 4 t^{5}\right)^{4}.

a

Determine the expression for h' \left( t \right).

b

Hence, find the exact values of t at which gradient of h \left( t \right) is zero.

14

Consider the function y = \left( 3 x + 2\right) \sqrt{3 + 8 x}.

a

Differentiate the function.

b

Hence, determine the value of x at which the gradient of the function is zero.

15

Consider the function f\left(x\right)=e^{5x}\left(e^x+e^{-x}\right).

a

Find f' \left( x \right).

b

Find f' \left( 0 \right).

16

Explain why the gradient of the function y = \sin ^{2}\left( 5 x\right) + \cos ^{2}\left( 5 x\right) is equal to 0 for all x.

17

If f \left( x \right) = \dfrac{\cos ^{2}\left(x\right)}{1 + \sin x}, evaluate f' \left( \dfrac{\pi}{4} \right).

18

If f \left( t \right) = \sqrt{1 + \sin ^{2}\left(t\right)}, evaluate f' \left( \dfrac{\pi}{2} \right).

19

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

b

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

20

Consider the curve y = \left(x - 8\right)^{2} \left( 2 x + 5\right). Find the equation of the tangents to the curve at the x-intercepts of the graph.

21

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}.

c

y = x \cos x at \left(\dfrac{\pi}{2}, 0\right).

d

y = - 5 x \sin x at \left(\pi, 0\right).

Sign up to access Worksheet
Get full access to our content with a Mathspace account
Book a Demo
Sign up for free

Outcomes

U34.AoS3.3

derivatives of f(x) +/- g(x), f(x) x g(x), f(x)/g(x) and (𝑓 ∘ 𝑔)(𝑥) where f and g are polynomial functions, exponential, circular, logarithmic or power functions and transformations or simple combinations of these functions

U34.AoS3.17

apply the product, chain and quotient rules for differentiation to simple combinations of functions by hand

U34.AoS3.16

find derivatives of polynomial functions and power functions, functions of the form f(ax+b) where f is x^n, sine, cosine; tangent, e^x, or log x base e and simple linear combinations of these, using pattern recognition, or by hand

U34.AoS3.18

find derivatives of basic and more complicated functions and apply differentiation to curve sketching and optimisation problems

What is Mathspace

About Mathspace