iGCSE (2021 Edition)

# 11.05 Chain rule

Worksheet
Chain rule
1

Given that y = u^{3} and u = 2 x + 3, define y in terms of x.

2

Redefine the following functions as composite functions f \left( u \right) and u \left( x \right), where u \left( x \right) is a polynomial:

a

f \left( x \right) = \left( 5 x^{3} - 4 x^{2} + 3 x - 5\right)^{7}

b

f \left( x \right) = \sqrt[4]{ 2 x^{2} + 2 x + 3}

c

f \left( x \right) = \dfrac{1}{\left( 4 x^{2} - 3 x + 5\right)^{3}}

3

To differentiate the function y = \left( 2 x^{4} + 6\right)^{5} using the substitution u = 2 x^{4} + 6:

a

Find \dfrac{d u}{d x}.

b

Express y as a function of u.

c

Find \dfrac{d y}{d u}.

d

Hence, find \dfrac{d y}{d x}.

4

For each of the following:

i

Find \dfrac{dy}{du}.

ii

Find \dfrac{du}{dx}.

iii

Hence, find \dfrac{dy}{dx}.

a

y = \left(x + 5\right)^{5}, where y = u^{5} and u = x + 5.

b

y = \left( 4 x + 3\right)^{ - 1 }, where y = u^{ - 1 } and u = 4 x + 3.

c

y = \sqrt{5 + x^{2}}, where y = \sqrt{u} and u = 5 + x^{2}.

d

y = \sqrt[3]{\dfrac{15}{x}}, where y = \sqrt[3]{u} and u = \dfrac{15}{x}.

5

Differentiate y = \left( 5 x - 7\right)^{2}:

a

By expanding the brackets first.

b

By using the chain rule.

6

Find the derivative of y = \dfrac{1}{x^{2} + 5}. Give your answer in positive index form.

7

Find the derivative of y = \sqrt{ 8 x + 5} using the chain rule. Give your answer in surd form.

8

Differentiate the following using the chain rule:

a

y = \left( 4 x + 3\right)^{9}

b

y = \left( 2 t^{7} + 8 t^{3} + 3 t + 5\right)^{ - 4 }

c

y = \left(x^{2} + x^{ - 3 }\right)^{3}

d

y = \sqrt[3]{x^{2} - 5 x}

e

y = \left( 3 x^{2} - 4 x + 2\right)^{4}

f

y = - 3 \left( 3 x + 4\right)^{10}

g

y = \dfrac{\left( 9 x + 7\right)^{\frac{4}{3}}}{3}

h

y = \dfrac{1}{\left(x + 6\right)^{5}}

i

y = \dfrac{1}{x^{4} - 4 x^{3} + 5 x}

j

y = \dfrac{2}{\sqrt{1 + x}}

k

y = \sqrt[5]{\left( 4 x + 1\right)^{6}}

l

y = - 4 \left( \dfrac{1}{3} x + 1\right)^{ - 6 }

m

y = 5 \sqrt{4 - \dfrac{1}{3} x}

n

y = \dfrac{2}{1 - x \sqrt{5}}

o

y = \left(\sqrt{x} + \dfrac{1}{\sqrt{x}}\right)^{8}

p
y = \dfrac{1}{\sqrt[3]{\left(9 - x\right)^{4}}}
q

y = - 3 \left(x + \dfrac{1}{x}\right)^{5}

9

Consider the function g \left( x \right) = \left(3 - x^{5}\right)^{4}. Use the chain rule to evaluate g' \left( - 1 \right).

10

Consider the function g \left( x \right) = \sqrt{7 - 3 x^{2}}.

a

Find g'\left(1\right).

b

Is the function increasing or decreasing at x = 1?

c

Find the range of values of x for which the function is increasing or decreasing.

11

Consider the function y = \sqrt{3 - 2 x}.

a

Find the derivative of the function.

b

To find where the tangent would be horizontal, what equation would we need to solve?

c

Which of the following x-values has a tangent that is closest to horizontal?

A

- \dfrac{3}{2}

B

10

C

- 10

D

\dfrac{3}{2}

d

Describe the behaviour of the gradient of the tangent as x decreases.

12

Find the gradient of f \left( x \right) = \left(x - 5\right)^{3} at the point \left(8, 27\right).

13

Find the gradient of the tangent to the curve y = \left( 3 x - 1\right)^{3} at the point \left(1, 8\right).

14

Find f' \left( 2 \right) for the function f \left( x \right) = 2 \left(x^{2} - 7\right)^{5}.

15

Find the x-coordinate of the point at which f \left( x \right) = \left(x - 2\right)^{2} has a gradient of 6.

16

Find the x-coordinate(s) of the point(s) at which f \left( x \right) = \left(x + 2\right)^{3} has a gradient of 48.

17

Find the values of x where the tangent of y = \left(x^{2} - 1\right)^{3} is horizontal.

18

Find the values of x where the derivative of y = \left( 2 x + x^{2}\right)^{5} is equal to zero.

19

The curve y = \sqrt{x - 3} has a tangent with a gradient of \dfrac{1}{2} at the point P. Find the coordinates of P.

20

For the function f \left( x \right) = \dfrac{1}{2 x - 7}, find the values of x where f' \left( x \right) = - \dfrac{2}{25}.

21

Consider the semicircle defined as y = \sqrt{169 - x^{2}}.

a

Find the derivative of the function.

b

Sketch the graph of the semicircle.

c

Find the equation of the tangent at the point \left(5, 12\right).

22

Find the equation of the tangent to y = \left( 2 x + 1\right)^{4} at the point where x = - 1.

23

Find the equation of the tangent to y = \left( 2 x - 1\right)^{8} at the point where x = 1.

24

Find the equation of the normal to y = \left( 3 x - 4\right)^{3} at the point \left(1, -1\right).

25

Find the equation of the normal to y = \left(x^{2} + 1\right)^{4} at the point where x = 1.

26

Consider the curve f \left( x \right) = \dfrac{1}{2 x + 3} at the point \left( - 1 , 1\right).

a

Find the equation of the tangent at the point.

b

Find the equation of the normal at the point.