How to differentiate — the rules with examples

How do you differentiate?

The derivative of a function measures its rate of change — the gradient of the tangent to the graph at each point. To differentiate you do not need limits: almost every question at AS and A level is solved with a handful of fixed rules. We go through them one at a time, including the one students ask about most — differentiating an expression in brackets. This is A level content; at GCSE you estimate a gradient by drawing a tangent.

The power rule — the central rule

If f(x) = xⁿ then the derivative is f′(x) = n·xⁿ⁻¹

The derivative at a point is the gradient of the tangent to the graph at that point — the steeper the graph, the larger the derivative.

Worked examples, step by step

The power rule

Find the derivative of f(x) = x³

By the power rule, the index comes down in front of the x and the new index is one less
f′(x) = 3·x²
Check at a point: at x = 2 the gradient is 3 × 4 = 12

Differentiating a sum and a constant multiple

Find the derivative of f(x) = 5x² + 3x − 7

Differentiate term by term (the derivative of a sum is the sum of the derivatives)
The derivative of 5x² is 5 × 2x = 10x
The derivative of 3x is 3, and the derivative of the constant −7 is 0
Answer: f′(x) = 10x + 3

Differentiating a bracket — the chain rule

Find the derivative of f(x) = (2x + 1)⁴

With a power on a bracket, differentiate "outside in": first the power, then multiply by the derivative of what is inside the bracket
Differentiating the power: 4·(2x + 1)³
The derivative of the inside, 2x + 1, is 2
Multiply: f′(x) = 4·(2x + 1)³ × 2 = 8·(2x + 1)³

The gradient of a tangent at a point

What is the gradient of the tangent to f(x) = x² at the point where x = 3?

Differentiate: f′(x) = 2x
Substitute the point: f′(3) = 2 × 3 = 6
Answer: the gradient of the tangent at that point is 6

Now try it yourself

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Frequently asked questions

What does the derivative actually tell you?

It measures how fast the function is changing: a positive derivative means the function is increasing, a negative one means decreasing, and zero means a "flat" point (a maximum, a minimum or a point of inflection). Geometrically it is the gradient of the tangent to the graph.

When do you need the chain rule?

Whenever there is a "function inside a function" — a power on a bracket, the root of an expression, the sine of an expression and so on. Differentiate the outer function, then multiply by the derivative of the inner one.

What is the derivative of a constant?

Zero. A constant function does not change, so its rate of change — its derivative — is 0 at every point.

How do you differentiate a product or a quotient of two functions?

By the product rule: (f·g)′ = f′·g + f·g′. For a quotient the derivative is (f′·g − f·g′) divided by g². Both formulae, with examples, are on the formulae page.

✍️ Written by the MathsUK teamChecked against the National Curriculum and GCSE specificationsLast updated: 13.09.2026