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

Gradients of curves and differentiation from first principles

70 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/a-level-differentiation-from-first-principles/

Date

Do Now

Quick questions to start.

A1
Needed today
Expand and simplify $(x + h)^{3} - x^{3}$ .
[2 marks]
Answer
A2
Last lesson
A particle moves in a straight line from $(2 i + 3 j)$ m to $(8 i - 5 j)$ m in $2$ seconds. Find its speed.
[2 marks]
Answer $\dots\dots m s^{- 1}$
A3
From chapter 5
Find the gradient of the straight line through $(- 2, 7)$ and $(4, - 11)$ .
[2 marks]
Answer
A4
From chapter 2
$f (x) = 2 x^{2} - 5 x$ . Find $f (3 + h)$ , simplified.
[2 marks]
Answer $f (3 + h) = \dots\dots$

Example, then your turn

A-Level

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

Example 1

$A (2, 8)$ lies on $y = x^{2} + 2 x$ . The tangent at $A$ passes through $(1, 2)$ . (a) Find the gradient of the curve at $A$ . (b) Find the gradient of the chord $A P$ when $P$ has $x$ -coordinate $3$ , then $2.1$ , then $2 + h$ . (c) Comment.

Not drawn accurately

Hint The gradient of a curve at a point is the gradient of the tangent there. A chord's gradient is

\frac{y _{2} - y _{1}}{x _{2} - x _{1}}

B1
$A (3, - 3)$ lies on $y = x^{2} - 4 x$ . The tangent at $A$ passes through $(5, 1)$ . (a) Find the gradient of the curve at $A$ . (b) Find the gradient of the chord $A P$ when $P$ has $x$ -coordinate $4$ , then $3.1$ , then $3 + h$ .
[5 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 2

The point $A (2, 10)$ lies on the curve $y = f (x)$ , where $f (x) = 3 x^{2} - x$ . At $A$ the curve has gradient $g$ . (a) Show that $g = h \to 0 lim (11 + 3 h)$ . (b) Deduce the value of $g$ .

Not drawn accurately

Hint

g = h \to 0 lim \frac{f ( 2 + h ) - f ( 2 )}{h}

. Expand

f (2 + h)

carefully: square the whole bracket.

B2
The point $A (2, 12)$ lies on the curve $y = x^{3} + 2 x$ . The point $B$ also lies on the curve and has $x$ -coordinate $2 + h$ . (a) Show that the gradient of $A B$ is $h^{2} + 6 h + 14$ . (b) Deduce the gradient of the curve at $A$ .
[5 marks]
Answer(b) $\dots\dots$

Example 3

Prove, from first principles, that the derivative of $x^{3} - 4 x$ is $3 x^{2} - 4$ .

Hint

f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h}

. Every term without an

h

cancels; then divide by

h

B3
Prove, from first principles, that the derivative of $2 x^{2} + 7 x - 1$ is $4 x + 7$ .
[5 marks]

Practice

A-Level

C1
$A (4, 2)$ lies on $y = x$ . Find, to $4$ d.p., the gradients of the chords from $A$ to $P (4.41, 2.1)$ and from $A$ to $Q (4.0401, 2.01)$ . Hence estimate the gradient of the curve at $A$ .
[3 marks]
Answer $A P$ : $\dots\dots$ $A Q$ : $\dots\dots$ estimate $\dots\dots$

C2
$A (- 2, 7)$ lies on $y = x^{2} + 3$ . $B$ is on the curve with $x$ -coordinate $- 2 + h$ . Show that the gradient of $A B$ is $h - 4$ . Find this gradient when $h = 0.1$ and when $h = - 0.1$ , and state the gradient of the curve at $A$ .
[4 marks]
Not drawn accurately
Answer $h = 0.1$ : $\dots\dots$ $h = - 0.1$ : $\dots\dots$ at $A$ : $\dots\dots$
C3
The points $(x, y)$ and $(x + δ x, y + δ y)$ lie on the curve $y = 5 - 2 x^{2}$ . Show that $δ y = - 4 x δ x - 2 (δ x)^{2}$ , and hence find $\frac{d y}{d x}$ .
[4 marks]
Answer $\frac{d y}{d x} = \dots\dots$

Exam-style questions

Edexcel exam style

D1
$f (x) = 2 x^{2} - 3 x + 4$ .
(a)
Prove, from first principles, that $f^{'} (x) = 4 x - 3$ .
[4 marks]
(b)
The points $P$ and $Q$ on the curve $y = f (x)$ have $x$ -coordinates $0$ and $4$ . Find the gradient of the chord $P Q$ .
[1 mark]
Answer
(c)
Hence find the coordinates of the point $R$ on the curve at which the gradient of the curve is equal to the gradient of $P Q$ .
[3 marks]
Answer $R (\dots\dots, \dots\dots)$
Total for D1: 8 marks

D2
Kai finds the gradient of $y = x^{3} + 5$ at $A (3, 32)$ . His working is: $\frac{( 3 + h ) ^{3} + 5 - 32}{h} = \frac{27 + h ^{3} - 27}{h} = h^{2}$ , which tends to $0$ . He says the gradient of the curve at $A$ is $0$ .
Kai is wrong.
(a)
Describe the mistake in Kai's working.
[2 marks]
(b)
Hence find the correct gradient of the chord, in terms of $h$ , and the gradient of the curve at $A$ .
[4 marks]
Answerchord $\dots\dots$ at $A$ : $\dots\dots$
Total for D2: 6 marks
D3
The curve $C$ has equation $y = x^{2} + k x$ , where $k$ is a constant. The point $P$ on $C$ has $x$ -coordinate $3$ .
(a)
Show that the gradient of the chord from $P$ to the point on $C$ with $x$ -coordinate $3 + h$ is $6 + k + h$ .
[3 marks]
(b)
The gradient of $C$ at $P$ is $11$ . Hence find the value of $k$ .
[2 marks]
Answer $k = \dots\dots$
(c)
Given that $\frac{d y}{d x} = 2 x + k$ , find the coordinates of the point on $C$ at which the gradient is $0$ .
[3 marks]
Answer $(\dots\dots, \dots\dots)$
Total for D3: 8 marks
D4
$f (x) = x^{3} - 3 x^{2} + 4$ .
(a)
Prove, from first principles, that $f^{'} (x) = 3 x^{2} - 6 x$ .
[5 marks]
(b)
Hence find the coordinates of the points on the curve $y = f (x)$ at which the gradient is $9$ .
[3 marks]
Answer $(\dots\dots, \dots\dots)$ and $(\dots\dots, \dots\dots)$
Total for D4: 8 marks

Extension

Stretch

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

E1
Prove, from first principles, that the derivative of $\frac{1}{x}$ is $- \frac{1}{x ^{2}}$ , for $x \neq = 0$ .
[6 marks]
Hint 1 · What to try
Write the gradient of the chord as $\frac{1}{h} (\frac{1}{x + h} - \frac{1}{x})$ .
Hint 2 · The first line
Put the two fractions over the common denominator $x (x + h)$ : the numerator is $- h$ .
Hint 3 · The full method
Cancel the $h$ , then let $h \to 0$ in what is left.

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use chords to estimate a gradient.
I can find a gradient from the definition.
I can prove a derivative from first principles.
I can find where the gradient takes a value.

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