A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 9: Differentiation · Lesson 2

The chain rule

114 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/a-level-the-chain-rule/

Date

Do Now

Quick questions to start.

A1
Needed today
Write $3 x - 1$ and $\frac{4}{( 2 x + 5 ) ^{3}}$ in the form $a (p x + q)^{n}$ .
[2 marks]
Answer
A2
Last lesson
Differentiate $y = 6 e^{4 x} - 2 ln (3 x) + 5 sin 2 x$ .
[3 marks]
Answer
A3
Common slip
Sam says the derivative of $(2 x + 1)^{5}$ is $5 (2 x + 1)^{4}$ . What has he left out? Give the correct derivative.
[2 marks]
A4
Needed today
Given $x = y^{3} + 2 y$ , find $\frac{d x}{d y}$ and its value when $y = 1$ .
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Given $y = (3 x^{2} - 5)^{4}$ , find $\frac{d y}{d x}$ .

Hint Let u be the bracket. Then y = u to the 4. Multiply the derivative of u to the 4 by the derivative of u.

B1
Given $y = (2 - 5 x^{3})^{6}$ , find $\frac{d y}{d x}$ and the gradient of the curve at the point where $x = 1$ .
[4 marks]
Answer

Example 2

Find the gradient of the curve $y = 4 x^{2} + 9$ at the point where $x = 2$ .

Hint Write the root as a power of one half. Bring the half down, take one from the power, and multiply by the derivative of the inside.

B2
The curve $C$ has equation $y = \frac{6}{( x ^{2} + 3 ) ^{2}}$ . Find an equation of the tangent to $C$ at the point where $x = 1$ , in the form $y = m x + c$ .
[5 marks]
Answer

Example 3

Differentiate (a) $y = e^{x^{2} - 3 x}$ (b) $y = ln (cos x)$ .

Hint For e to a function, multiply by the derivative of the power. For ln of a function, divide the derivative of the inside by the inside.

B3
Differentiate (a) $y = ln (5 - x^{2})$ (b) $y = cos^{4} (2 x)$ .
[5 marks]
Answer

Example 4

The curve $C$ has equation $x = y^{2} - 4 y$ . Find the gradient of $C$ at the point $(- 3, 1)$ .

Hint Differentiate x with respect to y, then turn it upside down.

B4
The curve $C$ has equation $x = 2 y^{3} + y - 1$ . Find an equation of the tangent to $C$ at the point where $y = 1$ , in the form $y = m x + c$ .
[5 marks]
Answer

Example 5

Given $x = e^{2 y}$ , find $\frac{d y}{d x}$ in terms of $x$ .

Hint Find dx/dy, write it in terms of x, then take 1 over it.

B5
Given $x = (y + 2)^{3}$ , find $\frac{d y}{d x}$ in terms of $x$ .
[4 marks]
Answer

Practice

A-Level

C1
Kai writes $\frac{d}{d x} (4 x - 3)^{- 2} = - 8 (4 x - 3)^{- 1}$ and $\frac{d}{d x} x^{2} + 1 = \frac{1}{2 x ^{2} + 1}$ . Find his mistake in each and give the correct derivatives.
[4 marks]
Answer

C2
Find the coordinates of the stationary points on the curve $y = (x^{2} - 4)^{3}$ and decide the nature of each.
[6 marks]
Answer
C3
The curve $x = y^{2} + 2 y$ passes through two points with $x = 3$ . Find the gradient of the curve at each of them.
[4 marks]
Answer
C4
Show that the curve $y = ln (x^{2} + 2 x + 5)$ has exactly one stationary point. Find its exact coordinates and its nature.
[5 marks]
Answer
C5
Lena says that $\frac{d}{d x} (e^{x^{2}}) = x^{2} e^{x^{2} - 1}$ . Explain why she is wrong, and find the exact gradient of $y = e^{x^{2}}$ at $x = 1$ .
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
The curve $C$ has equation $y = \frac{4}{x ^{2} - 6 x + 13}$ .
(a)
Show that $\frac{d y}{d x} = \frac{8 ( 3 - x )}{( x ^{2} - 6 x + 13 ) ^{2}}$ .
[3 marks]
(b)
Find the coordinates of the stationary point of $C$ and decide its nature.
[4 marks]
Answer
(c)
Find an equation of the normal to $C$ at the point where $x = 1$ , in the form $y = m x + c$ .
[4 marks]
Answer
Total for D1: 11 marks

D2
Omar wants the tangent to the curve $x = y^{3} - 3 y$ at the point $(2, 2)$ . He writes: " $\frac{d y}{d x} = 3 y^{2} - 3 = 9$ at the point, so the tangent is $y - 2 = 9 (x - 2)$ ."
Omar is wrong.
(a)
Explain his mistake.
[2 marks]
(b)
Find the correct equation of the tangent in the form $y = m x + c$ .
[3 marks]
Answer
(c)
Find the coordinates of the points on the curve where the tangent is parallel to the $y$ -axis.
[4 marks]
Answer
Total for D2: 9 marks
D3
Show that the gradient of the curve $y = ln (x^{2} + 1)$ is never greater than $1$ . Find the point where the gradient equals $1$ , and an equation of the tangent at that point.
[9 marks]
Answer
D4
The number of birds, $P$ , on an island $t$ years after they arrive is modelled by $P = \frac{800}{1 + 3 e ^{- 0.4 t}}$ , $t \geq 0$ .
(a)
Find the number of birds when they arrive.
[2 marks]
Answer
(b)
Show that $\frac{d P}{d t} = \frac{960 e ^{- 0.4 t}}{( 1 + 3 e ^{- 0.4 t} ) ^{2}}$ .
[3 marks]
(c)
Find the rate at which the number of birds is growing when $t = 5$ .
[2 marks]
Answer
(d)
Find the exact time at which there are $400$ birds, and give it to $3$ significant figures.
[3 marks]
Answer
(e)
State the number of birds the model gives in the long term, and give one reason why the model may not be right for large values of t.
[2 marks]
Total for D4: 12 marks
D5
The curve $C$ has equation $x = e^{2 y} - 4 y$ .
(a)
Find $\frac{d x}{d y}$ .
[2 marks]
Answer
(b)
Find the exact coordinates of the point on $C$ where the tangent is parallel to the $y$ -axis.
[3 marks]
Answer
(c)
Hence, or otherwise, find an equation of the normal to $C$ at the point where $y = 0$ .
[4 marks]
Answer
Total for D5: 9 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The curve $C$ has equation $y = ln (x^{2} + a x + 9)$ , where $a$ is a constant. Find the set of values of $a$ for which $C$ has a stationary point. Find the values of $a$ for which this stationary point has $y$ -coordinate $ln 5$ .
[9 marks]
Hint 1 · What to try
Differentiate with the chain rule. Where can the top of your fraction be zero?
Hint 2 · The first line
A logarithm only exists when the inside is positive. Is the curve defined at the $x$ you found?
Hint 3 · The full method
Put $x = - \frac{a}{2}$ into $x^{2} + a x + 9$ . When is this positive, and when does it equal $5$ ?
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can differentiate a function of a function, such as a power of a bracket, e to a function or ln of a function.
I can use the chain rule more than once when one function sits inside another inside another.
I can use the fact that the gradient is 1 divided by the rate of change of x with respect to y.
I can use these to find tangents, normals and stationary points.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers