A-Level Maths / Pure Mathematics / Differentiation

Differentiating Polynomials

First principles, differentiating x^n, gradient functions, rates of change.

Pure Mathematics AS 40 min

Learning Objectives

Understand differentiation as finding the gradient function
Differentiate powers of x using the power rule
Differentiate sums and differences of terms
Rewrite expressions (roots, fractions, products) before differentiating
Find the gradient at a specific point by substituting into the derivative

Key Formulae

\frac{d}{dx}(x^n) = nx^{n-1}

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

\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)

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

Prior Knowledge Check

Answer at least 3 of 3 correctly to complete this section.

Q1. Rewrite

\sqrt{x}

as a power of

x

Q2. Simplify

\dfrac{x^3}{x}

Q3. Rewrite

\dfrac{1}{x^2}

using a negative index.

Why This Matters

How fast is a car going at a particular instant? How quickly is a population growing right now? At what rate is temperature changing this second?

All of these are rates of change — and differentiation is the tool that answers them. In geometry, the derivative gives you the gradient of a curve at any point. In the real world, it gives you the instantaneous rate of change of anything that varies.

Differentiation is the single most-used technique in A-Level Maths. It appears in curve sketching, optimisation, kinematics, and practically every applied topic you will meet. Master the power rule here, and the rest of calculus builds on it.

1/5

What is Differentiation?

2/5

The Power Rule

$y$	$\frac{dy}{dx}$	Note
$x^5$	$5x^4$	Bring down 5, reduce power to 4
$3x^4$	$12x^3$	$3 \times 4 = 12$ , power drops to 3
$7x$	$7$	Think of $7x^1$ : power becomes $x^0 = 1$
$5$	$0$	Constants differentiate to zero
$-2x^3$	$-6x^2$	The negative carries through

3/5

Rewriting Before Differentiating

Original form	Rewrite as	Derivative
$\sqrt{x}$	$x^{1/2}$	$\frac{1}{2}x^{-1/2}$
$\frac{1}{x}$	$x^{-1}$	$-x^{-2}$
$\frac{5}{x^3}$	$5x^{-3}$	$-15x^{-4}$
$x\sqrt{x}$	$x^{3/2}$	$\frac{3}{2}x^{1/2}$
$\frac{2}{\sqrt{x}}$	$2x^{-1/2}$	$-x^{-3/2}$

4/5

Finding Gradients at Points

5/5

Exam Practice

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Exam Tips

Rewrite roots and fractions as powers before differentiating — e.g. √x = x^{1/2}, 1/x = x^{-1}
Expand brackets before differentiating if there's no product or chain rule needed
The derivative of a constant is 0
Write dy/dx clearly — don't mix up y and dy/dx in your working
Check your answer by substituting a value — the gradient should match the curve's slope