Topics
2. Differentiation
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Year 12

2.07 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 =\cos \left(\cos x \right)
j
y = \sqrt{\cos 4 x}
k
y=\dfrac{\cos x - \sin x}{\cos x + \sin x}
l
y = \dfrac{\cos \left( 4 x - \dfrac{10 \pi}{11}\right)}{\left(x + 3\right)^{2}}
4

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^{ - x } \sin 4 x
h
y = \dfrac{e^{ x }} {\cos x}
i
y = 4 \sin \left(\dfrac{x}{5}\right) - 6 e^{ 2 x} + x^{ - 8 }
j
y = \left(e^{ - 5 x^{2} } + \cos x\right)^{5}
k
y = \dfrac{e^{ - 0.2 x}}{\sin \left( \dfrac{\pi}{4} x\right) - x^{4}}
l
y = \left(\cos x + \sin x\right) e^{ 6 x}
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 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'.

Gradients and tangents
7

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

8

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

9

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.

10

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.

11

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.

12

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).

13

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.

14

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

15

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

16

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}.

17

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.

18

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).

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Outcomes

ACMMM105

understand the notion of composition of functions and use the chain rule for determining the derivatives of composite functions

ACMMM106

apply the product, quotient and chain rule to differentiate functions such as xe^x, tan⁡x,1/x^n, x sin⁡x, e^(−x)sin⁡x and f(ax+b)

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