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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

LearnYMA01WMA11 · Pure Mathematics 1DifferentiationDifferentiating powers of x

Differentiation · Subtopic 2 of 4

Differentiating Powers of x

The derivative $\dfrac{dy}{dx}$ is the gradient of a curve, found as the limit of the gradients of shorter and shorter chords. For powers of $x$ there is a one-line rule: bring the power down and reduce it by one. Rewrite roots and fractions as powers first.

Edexcel YMA01 WMA11≈ 30 minRevision 5 · updated 2026-10-27

Practise this topicTimed test Ask Salam Tutor

Section 1 of 14

Learning objectives

Understand the derivative as the limit of chord gradients.
Differentiate $x^n$ for any rational $n$ .
Rewrite roots and fractions as powers before differentiating.
Expand or divide before differentiating products and quotients.

Pause and think

On $y=x^2$ , the chord from $x=2$ to $x=2+h$ has gradient $4+h$ . What happens as $h$ gets smaller and smaller?

Section 2 of 14

Core idea

Start hereSpeed at an instant

A speedometer shows your speed right now. But speed is distance ÷ time, and in an instant both are zero. Calculus resolves this by taking smaller and smaller intervals and watching the ratio settle. That limit is the derivative.

The big idea

$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}h$ . For powers: $\frac{d}{dx}x^n=nx^{n-1}$ ; constants multiply through; terms differentiate separately.

1. First principles

Chords from (1, 1) on y = x² approach the tangent

h = 2.00

Gradient of the chord

=\dfrac{(1+h)^2-1}{h}=2+h

. As

h\to0

this becomes

2

: the gradient of the curve at

x=1

f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}

$h$: small step in $x$ .
$\frac{f(x+h)-f(x)}{h}$: gradient of a chord.

\frac{d}{dx}x^3=3x^2

$(x+h)^3=x^3+3x^2h+3xh^2+h^3$
Binomial expansion.
$\frac{(x+h)^3-x^3}h=\frac{3x^2h+3xh^2+h^3}h$
$x^3$ cancels.
$=3x^2+3xh+h^2$
Divide by $h\neq0$ .
$\to3x^2\text{ as }h\to0$
Terms containing $h$ vanish.

The same pattern gives

nx^{n-1}

for any power.

2. Using the rule

Worked example $y=2x^4-\frac3x+\sqrt x$

Find $\frac{dy}{dx}$ .

Goal: Derivative.
Strategy: Rewrite every term as a power of $x$ first.

$y=2x^4-3x^{-1}+x^{1/2}$
Powers.
$\frac{dy}{dx}=8x^3+3x^{-2}+\tfrac12x^{-1/2}$
$nx^{n-1}$ each term.
$=8x^3+\frac3{x^2}+\frac1{2\sqrt x}$
Tidy.

Answer

8x^3+\frac3{x^2}+\frac1{2\sqrt x}

✓ Check

-3x^{-1}\to(-1)(-3)x^{-2}=+3x^{-2}

: two negatives make positive ✓.

Common misconceptionFractions with x underneath

\frac1{x^2}

x^{-2}

; you cannot differentiate the denominator alone.

Pause and think

Differentiate $5$ .

Summary

Limit of chord gradients.
$nx^{n-1}$ after rewriting as powers.

Derivative: Gradient function.
First principles: Differentiating from the limit definition.

HistoryThe calculus dispute. Isaac Newton and Gottfried Leibniz invented calculus independently in the 1660s–1680s. A bitter argument over who came first lasted decades. We use Leibniz's notation

\dfrac{dy}{dx}

and Newton's dot notation still appears in physics.

Fun factInfinitely small? Early calculus used "infinitesimals", amounts smaller than any number yet not zero. Bishop Berkeley mocked them in 1734 as "the ghosts of departed quantities". It took until the 1800s for limits to put calculus on firm ground.

Section 3 of 14

Key formulae

Derivative as a limit

f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}h

Power rule

\frac{d}{dx}\left(ax^n\right)=anx^{n-1}

Constants

\frac{d}{dx}(c)=0

TrickRewrite, then differentiate. Before differentiating, make every term look like

ax^n

. A line of rewriting ("

y=6x^{\frac12}+2x^{-2}

") prevents almost every power-rule mistake and often earns a mark on its own.

Section 4 of 14

Worked examples

Example 1 · Roots and fractions0 / 2 steps

Given

y=6\sqrt x+\dfrac2{x^2}

, find

\dfrac{dy}{dx}

Example 2 · Divide first0 / 2 steps

Given

y=\dfrac{x^2+3}{\sqrt x}

, find

\dfrac{dy}{dx}

Section 5 of 14

Explore the graph

Chord to tangent

Loading the interactive graph…

Try this

The line is the chord from $x=a$ to $x=a+h$ on $y=x^2$ ; the value is its gradient $2a+h$ .
Slide $h$ towards $0$ . The chord turns into the tangent, and its gradient approaches $2a$ .
So the derivative of $x^2$ is $2x$ . Check with the power rule.

The derivative is the limit of the chord gradient as $h\to0$ .

In the real worldSpeedometers. A car's speed is the derivative of its distance with respect to time. Your speedometer is effectively measuring the gradient of a distance–time graph at each instant.

Section 6 of 14

Common mistakes

Incorrect

\dfrac{d}{dx}\sqrt x=\tfrac12x^{\frac32}

Correct

\tfrac12x^{-\frac12}

Reduce the power by

1

\tfrac12-1=-\tfrac12

IncorrectConstant term stays

CorrectIt differentiates to

0

A constant has zero gradient.

IncorrectDifferentiating a product factor by factor

CorrectExpand first

(2x+5)(x-1)

is not

2\times1

Incorrect

\dfrac{d}{dx}x^{-2}=-2x^{-1}

Correct

-2x^{-3}

-2-1=-3

Expectation

$\dfrac{d}{dx}\left(\dfrac{1}{x^2}\right)=\dfrac{1}{2x}$

Reality

Write it as $x^{-2}$ first: $-2x^{-3}$

Section 7 of 14

Exam tip

Exam tip"Simplest form" means no brackets. When asked for each term in its simplest form, give coefficients as simplified fractions and powers as single indices:

\tfrac32x^{\frac12}

, not

\tfrac{3}{2\sqrt{x^{-1}}}

Section 8 of 14

Guided practice

SA-IAL-P1-DF-G012 marksL1 FoundationCalculatorPower rule

Given

y=4x^3-5x^2+2

, find

\dfrac{dy}{dx}

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-IAL-P1-DF1-0011 markL1 FoundationNon-calculatorPower rule

Given

y=5x^4

, find

\dfrac{dy}{dx}

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-DF1-0022 marksL1 FoundationNon-calculatorPower rule

Given

y=\dfrac3x

, find

\dfrac{dy}{dx}

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-DF1-0032 marksL1 FoundationNon-calculatorPower rule

Given

y=x\sqrt x

, find

\dfrac{dy}{dx}

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-DF-0022 marksL1 FoundationCalculatorPower rule

Given

y=(2x+5)(x-1)

, find

\dfrac{dy}{dx}

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-DF-0013 marksL2 StandardCalculatorPower rule

Given

y=6\sqrt x+\dfrac{2}{x^2}

, find

\dfrac{dy}{dx}

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-DF1-0042 marksL2 StandardNon-calculatorFirst principles

y=x^2

, find the gradient of the chord from

x=3

x=3.01

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-DF-0102 marksL2 StandardCalculatorFirst principles

On the curve

y=x^2

, find the gradient of the chord from the point where

x=2

to the point where

x=2.1

. (As the second point moves towards the first, this tends to the gradient of the tangent,

4

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-IAL-P1-DF-0034 marksL3 ExamCalculatorPower rule

Given

y=\dfrac{x^2+3}{\sqrt x}

, find

\dfrac{dy}{dx}

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

$\dfrac{dy}{dx}$ is the limit of chord gradients.
$\dfrac{d}{dx}(ax^n)=anx^{n-1}$ ; constants give $0$ .
Rewrite roots and fractions as powers.
Expand products and divide quotients before differentiating.

Section 12 of 14

Quick quiz

Q1 · SA-IAL-P1-DF-Q011 markL1 FoundationCalculatorPower rule

\dfrac{d}{dx}\left(\dfrac1x\right)=

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-IAL-P1-DF-Q031 markL1 FoundationCalculatorPower rule

\dfrac{d}{dx}(7)=

Salam Academy Original Practice · Proposed mark scheme

Q3 · SA-IAL-P1-DF1-0051 markL1 FoundationCalculatorFirst principles

The gradient of the tangent to a curve at a point is

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: IAL Mathematics, unit WMA11 (Pure 1)
Where: P1 · 7.1–7.2 The derivative and differentiating xⁿ
Expected: Understand the gradient of a curve as the limit of the gradients of chords; differentiate xⁿ for rational n, and sums and differences of such terms, after simplifying.
Calculator: Calculators are permitted, but exact answers (surds, fractions, π) are expected where the question asks.
Typical style: "Given y = …, find dy/dx, giving each term in its simplest form." Expressions are often given with roots and fractions that must be rewritten first.
Open in: YMA01 · WMA11

Section 14 of 14

Past paper questions

3 questions (20 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

WMA11/01 · June 2025Question 25 marks
Differentiate a power curve and find where the gradient takes a given value.
Question paper, page 4 Mark scheme
WMA11/01 · January 2025Question 58 marks
Differentiate a curve and use a tangent condition to find a constant.
Question paper, page 14 Mark scheme
WMA11/01 · June 2023Question 47 marks
Write an expression using indices, then differentiate.
Question paper, page 8 Mark scheme

All YMA01 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel YMA01, 2018 specification (official specification).