A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 12: Differentiation · Lesson 2

Differentiating xⁿ and sums of terms

72 marks

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Quick questions to start.

A1
Needed today
Write $\frac{x ^{2} + 5}{x}$ as a sum of powers of $x$ .
[2 marks]
Answer
A2
Last lesson
$A (1, 3)$ lies on $y = x^{2} + 2$ . Find the gradient of the chord from $A$ to the point on the curve with $x$ -coordinate $1 + h$ .
[2 marks]
Answer
A3
From chapter 8
Find the coefficient of $x^{2}$ in the expansion of $(1 + 2 x)^{5}$ .
[2 marks]
Answer
A4
From chapter 2
Solve $3 x^{2} - 2 x - 8 = 0$ .
[2 marks]
Answer $x = \dots\dots$

Example, then your turn

A-Level

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

Example 1

Find $\frac{d y}{d x}$ when $y$ equals: (a) $5 x^{4}$ (b) $\frac{6}{x ^{2}}$ (c) $4 3 x$ (d) $\frac{x ^{5}}{2 x ^{2}}$

Hint Write each one as

a x^{n}

first. Then multiply by the power and take

1

off the power.

B1
Find $\frac{d y}{d x}$ when $y$ equals: (a) $3 x^{7}$ (b) $\frac{10}{x ^{5}}$ (c) $8 4 x$ (d) $\frac{x ^{2} \times x ^{4}}{3 x}$ (e) $16 x^{5}$
[5 marks]

Example 2

$y = 2 x^{3} - \frac{4}{x} + 3 x - 7$ , $x > 0$ . Find $\frac{d y}{d x}$ , and find the gradient of the curve at the point $(4, 126)$ .

Hint Differentiate one term at a time. A constant term has gradient

0

B2
$y = x^{3} + \frac{16}{x ^{2}} - 4 x$ , $x > 0$ . Find $\frac{d y}{d x}$ , and find the gradient of the curve at the point $A (4, 57)$ .
[5 marks]
Answergradient at $A$ : $\dots\dots$

Example 3

$f (x) = \frac{( 2 x - 1 ) ^{2}}{x}$ , $x \neq = 0$ . (a) Write $f (x)$ in the form $A x + B + C x^{- 1}$ . (b) Find $f^{'} (x)$ . (c) Find the coordinates of the points on $y = f (x)$ where the gradient is $0$ .

Hint Expand the bracket, then divide every term by

x

. You cannot differentiate the top and the bottom separately.

B3
$f (x) = \frac{( 3 x + 2 ) ^{2}}{x}$ , $x \neq = 0$ . Find $f^{'} (x)$ , and find the coordinates of the points on $y = f (x)$ where the gradient is $5$ .
[6 marks]
Answer $(\dots, \dots)$ , $(\dots, \dots)$

Practice

A-Level

C1
$f (x) = 3 x^{2} - 7 x + 2$ . (a) Find the gradient of $y = f (x)$ at the point $(2, 0)$ . (b) Find the coordinates of the point on $y = f (x)$ at which the gradient is $11$ .
[4 marks]
Answer(a) $\dots\dots$ (b) $(\dots\dots, \dots\dots)$

C2
The curve $y = x^{2} - 3 x - 1$ meets the line $y = 2 x - 5$ at the points $A$ and $B$ . Find the gradient of the curve at $A$ and at $B$ .
[4 marks]
Not drawn accurately
Answerat $A$ : $\dots\dots$ at $B$ : $\dots\dots$
C3
Given that $9 y^{2} = 4 x^{5}$ and $y > 0$ , find $\frac{d y}{d x}$ . Hence find the gradient of the curve at the point where $x = 9$ .
[4 marks]
Answer $\frac{d y}{d x} = \dots\dots$ gradient $\dots\dots$

Exam-style questions

Edexcel exam style

D1
Priya differentiates $y = \frac{x ^{3} + 16}{2 x}$ , $x \neq = 0$ . She writes: "The top differentiates to $3 x^{2}$ and the bottom to $2$ , so $\frac{d y}{d x} = \frac{3 x ^{2}}{2}$ ."
Priya is wrong.
(a)
Explain the mistake in Priya's method.
[2 marks]
(b)
Find $\frac{d y}{d x}$ correctly.
[3 marks]
Answer $\frac{d y}{d x} = \dots\dots$
(c)
Hence find the coordinates of the point on the curve $y = \frac{x ^{3} + 16}{2 x}$ at which the gradient is $0$ .
[3 marks]
Answer $(\dots\dots, \dots\dots)$
Total for D1: 8 marks

D2
The curve $C$ has equation $y = a x^{2} + \frac{b}{x}$ , $x \neq = 0$ , where $a$ and $b$ are constants. The gradient of $C$ is $- 4$ at the point where $x = 1$ , and $6$ at the point where $x = 2$ .
(a)
Show that $2 a - b = - 4$ .
[2 marks]
(b)
Hence find the value of $a$ and the value of $b$ .
[3 marks]
Answer $a = \dots\dots$ $b = \dots\dots$
(c)
The point $P$ on $C$ has $x$ -coordinate $- 1$ . Find the coordinates of $P$ and the gradient of $C$ at $P$ .
[3 marks]
Answer $P (\dots\dots, \dots\dots)$ gradient $\dots\dots$
Total for D2: 8 marks
D3
$f (x) = \frac{k}{x} + 3 x$ , $x > 0$ , where $k$ is a constant.
(a)
Find $f^{'} (x)$ , in terms of $k$ and $x$ .
[2 marks]
Answer $f^{'} (x) = \dots\dots$
(b)
Given that $f^{'} (2) = 2$ , find the value of $k$ , giving your answer in the form $a 2$ , where $a$ is an integer.
[3 marks]
Answer $k = \dots\dots$
(c)
Hence find the exact value of $f (8)$ .
[2 marks]
Answer
Total for D3: 7 marks
D4
$f (x) = (1 + 4 x)^{5}$ .
(a)
Find the first three terms, in ascending powers of $x$ , of the binomial expansion of $f (x)$ , giving each term in its simplest form.
[3 marks]
Answer $\dots\dots + \dots\dots + \dots\dots$
(b)
Given that $x$ is small, so that $x^{2}$ and higher powers of $x$ in $f^{'} (x)$ can be ignored, hence find an approximation for $f^{'} (x)$ in the form $p + q x$ .
[2 marks]
Answer $f^{'} (x) \approx \dots\dots$
(c)
Use your answer to part (b) to estimate the gradient of $y = f (x)$ at the point where $x = 0.01$ .
[2 marks]
Answer
Total for D4: 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 $y = \frac{( x + 2 ) ^{3}}{x}$ , $x > 0$ . Show that there is exactly one point on $C$ at which the gradient is $0$ , and find its coordinates.
[6 marks]
Hint 1 · What to try
Expand $(x + 2)^{3}$ as you would $(u + 2)^{3}$ , with $u = x$ .
Hint 2 · The first line
Divide every term by $x$ and write each as a power of $x$ before you differentiate.
Hint 3 · The full method
Multiply $\frac{d y}{d x} = 0$ by $2 x^{2}$ and put $u = x$ . One root of the cubic in $u$ is a whole number.

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can differentiate any power of x.
I can rewrite roots and fractions as powers.
I can find the gradient at a point.
I can find where the gradient takes a value.

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