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2. Differentiation
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Year 12

2.07 Further differentiation

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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

3.2.4.1

select and apply the product rule, quotient rule and chain rule to differentiate functions; express derivatives in simplest and factorised form

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