A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 9: Differentiation · Lesson 3

The product and quotient rules

117 marks

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Date

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Quick questions to start.

A1
Last lesson
Differentiate $y = (3 x - 2)^{5}$ .
[2 marks]
Answer
A2
Last lesson
Differentiate $y = e^{x^{2} + 1}$ .
[2 marks]
Answer
A3
Needed today
Factorise fully $2 x (x + 3)^{4} + 4 (x + 3)^{3}$ .
[3 marks]
Answer
A4
Common slip
Dev says the derivative of $x^{2} sin x$ is $2 x cos x$ . Explain why he is wrong and give the correct derivative.
[2 marks]

Example, then your turn

A-Level

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

Example 1

Given $y = x^{3} e^{2 x}$ , find $\frac{d y}{d x}$ and write it in factorised form.

Hint Let u = x cubed and v = e to the 2x. The derivative is u times the derivative of v, plus v times the derivative of u.

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

Example 2

Given $y = (2 x + 1) (x - 3)^{4}$ , show that $\frac{d y}{d x} = 2 (x - 3)^{3} (5 x - 1)$ .

Hint Use the product rule, with the chain rule for the bracket to the power 4. Then take out the common factor 2 times the bracket cubed.

B2
Given $y = x (2 x - 1)^{5}$ , show that $\frac{d y}{d x} = (2 x - 1)^{4} (12 x - 1)$ , and find the $x$ -coordinates of the stationary points of the curve.
[5 marks]
Answer

Example 3

Given $y = \frac{2 x + 1}{x - 3}$ , find $\frac{d y}{d x}$ .

Hint With u on top and v underneath, the derivative is v times the derivative of u, minus u times the derivative of v, all over v squared. The order on top matters.

B3
The curve $C$ has equation $y = \frac{3 x - 2}{x ^{2} + 1}$ . 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 4

Find the exact coordinates of the stationary point on the curve $y = \frac{e ^{2 x}}{x}$ , $x \neq = 0$ .

Hint Use the quotient rule, take out the factor e to the 2x on top, then set the top equal to zero.

B4
Find the exact coordinates of the stationary point on the curve $y = \frac{ln x}{x ^{2}}$ , $x > 0$ .
[5 marks]
Answer

Example 5

Find the gradient of the curve $y = x^{2} cos x$ at the point where $x = π$ .

Hint Product rule, then put in x = pi. Remember sin pi is 0 and cos pi is minus 1.

B5
Find the $x$ -coordinates of the stationary points on the curve $y = e^{x} sin x$ for $0 \leq x \leq 2 π$ .
[4 marks]
Answer

Practice

A-Level

C1
Ravi differentiates $y = \frac{x ^{2} + 1}{x - 1}$ and writes $\frac{d y}{d x} = \frac{( x ^{2} + 1 ) - 2 x ( x - 1 )}{( x - 1 ) ^{2}}$ . Explain his mistake, and find the $x$ -coordinates of the stationary points of the curve.
[4 marks]
Answer

C2
Find the exact coordinates of the stationary points on the curve $y = x^{2} e^{- x}$ and decide the nature of each.
[6 marks]
Answer
C3
Find the set of values of $x$ for which $f (x) = \frac{x}{x ^{2} + 4}$ is increasing.
[4 marks]
Answer
C4
Find the exact coordinates of the stationary point on the curve $y = x ln x$ , $x > 0$ .
[5 marks]
Answer
C5
Given $y = (x - 1)^{2} (x + 2)^{3}$ , show that $\frac{d y}{d x} = (x - 1) (x + 2)^{2} (5 x + 1)$ , and find the $x$ -coordinates of the stationary points.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
The curve $C$ has equation $y = (x^{2} - 3) e^{x}$ .
(a)
Show that $\frac{d y}{d x} = (x^{2} + 2 x - 3) e^{x}$ .
[3 marks]
(b)
Find the exact coordinates of the stationary points of C.
[4 marks]
Answer
(c)
Find $\frac{d ^{2} y}{d x ^{2}}$ , and hence decide the nature of each stationary point.
[4 marks]
Answer
Total for D1: 11 marks

D2
Priya finds the stationary points of $y = \frac{x ^{2}}{x - 1}$ . She differentiates the top and the bottom separately: " $\frac{d y}{d x} = \frac{2 x}{1} = 2 x$ , so the only stationary point is $(0, 0)$ ."
Priya is wrong.
(a)
Find $\frac{d y}{d x}$ correctly, in its simplest form.
[3 marks]
Answer
(b)
Find the coordinates of both stationary points.
[3 marks]
Answer
(c)
Decide the nature of each stationary point.
[3 marks]
Total for D2: 9 marks
D3
The curve $C$ has equation $y = \frac{2 x - 1}{x ^{2} + 2}$ . Find the exact coordinates of the points on $C$ where the tangent is parallel to the $x$ -axis.
[6 marks]
Answer
D4
After a dose of a drug, the amount in a patient's blood, $C$ mg per litre, is modelled by $C = 50 t e^{- 0.2 t}$ , where $t$ is the time in hours.
(a)
Show that $\frac{d C}{d t} = 50 e^{- 0.2 t} (1 - 0.2 t)$ .
[3 marks]
(b)
Find the greatest amount of the drug in the blood, and the time at which it happens. Give the amount to 3 significant figures.
[3 marks]
Answer
(c)
Find the rate of change of $C$ when $t = 10$ , to 3 significant figures.
[2 marks]
Answer
(d)
Explain what the sign of your answer to part (c) means in this context.
[2 marks]
Total for D4: 10 marks
D5
The curve $C$ has equation $y = (x^{2} + 1) ln x$ , $x > 0$ .
(a)
Find $\frac{d y}{d x}$ .
[3 marks]
Answer
(b)
Find an equation of the normal to $C$ at the point where $x = 1$ .
[4 marks]
Answer
(c)
The normal meets the $y$ -axis at $A$ . Find the area of the triangle formed by the normal and the two axes.
[2 marks]
Answer
Total for D5: 9 marks
D6
The curve $C$ has equation $y = e^{2 x} cos x$ , $0 \leq x \leq \frac{π}{2}$ .
(a)
Show that at the stationary point $tan x = 2$ .
[3 marks]
(b)
Find the coordinates of the stationary point, to 3 significant figures.
[4 marks]
Answer
Total for D6: 7 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 $y = \frac{x ^{2} + k}{x - 1}$ , $x \neq = 1$ , where $k$ is a constant. Find the set of values of $k$ for which $C$ has two stationary points. When $k = 3$ , find the stationary points and decide the nature of each.
[9 marks]
Hint 1 · What to try
Differentiate with the quotient rule. The bottom of your answer is never negative, so where can the gradient be zero?
Hint 2 · The first line
The top is a quadratic in x. How many roots does it have, and when are there two? Could one of them be the excluded value x = 1?
Hint 3 · The full method
For $k = 3$ the top is $(x - 3) (x + 1)$ . Look at its sign just below and just above each root.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can differentiate a product with the product rule.
I can differentiate a quotient with the quotient rule, with the terms on top in the right order.
I can combine these with the chain rule and factorise the answer.
I can use the derivative to find tangents, normals and stationary points.

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