A level Maths differentiation

By The STEM-Tutor team · Updated October 2026

Differentiation finds rates of change and gradients. Here is everything A level Maths expects, from first principles to applications.

From first principles

The derivative is the limit of the gradient of a chord as it shrinks: f'(x) = lim (f(x + h) − f(x))/h as h → 0. You may be asked to prove simple results this way.

Standard results

• d/dx(xⁿ) = nxⁿ⁻¹
• d/dx(e^kx) = ke^kx and d/dx(ln x) = 1/x
• d/dx(sin kx) = k cos kx and d/dx(cos kx) = −k sin kx

Trigonometric results need angles in radians.

The three key rules

• Chain rule: dy/dx = dy/du × du/dx, for functions of functions.
• Product rule: d/dx(uv) = u(dv/dx) + v(du/dx).
• Quotient rule: d/dx(u/v) = (v(du/dx) − u(dv/dx))/v².

Further techniques

• Implicit differentiation for curves such as x² + y² = 25.
• Parametric differentiation: dy/dx = (dy/dt)/(dx/dt).
• Connected rates of change using the chain rule.

Applications

• Gradients of tangents and normals.
• Stationary points: set dy/dx = 0, then use the second derivative to classify them.
• Increasing and decreasing functions, and convex and concave curves.
• Optimisation problems, such as maximising area or minimising cost.

Common mistakes

• Using degrees in trigonometric derivatives.
• Forgetting the chain rule on the inner function.
• Not justifying the nature of a stationary point.

Why differentiation matters

Differentiation measures how fast one quantity changes compared with another. On a graph, it gives the gradient of the curve at any point. In real problems it tells you speed from position, the rate a population grows, or where a cost is smallest. Almost every A level paper uses it, often combined with other topics, so it is worth making completely secure.

Worked examples

Example 1: from first principles

Prove that the derivative of x² is 2x.

• The gradient of the chord from x to x + h is ((x + h)² − x²)/h = (2xh + h²)/h = 2x + h.
• As h → 0, this tends to 2x. Write the limit statement clearly; the proof marks depend on it.

Example 2: the chain rule

Differentiate y = (3x − 1)⁴.

• Let u = 3x − 1, so y = u⁴. dy/dx = 4u³ × 3 = 12(3x − 1)³.

Example 3: the product rule

Differentiate y = x² sin x.

• dy/dx = 2x sin x + x² cos x = x(2 sin x + x cos x).

Example 4: the quotient rule

Differentiate y = e^x/x.

• dy/dx = (x e^x − e^x)/x² = e^x(x − 1)/x².

Example 5: stationary points

Find and classify the stationary points of y = x³ − 6x² + 9x.

• dy/dx = 3x² − 12x + 9 = 3(x − 1)(x − 3), so x = 1 or x = 3.
• d²y/dx² = 6x − 12. At x = 1 it is −6, so a maximum at (1, 4). At x = 3 it is 6, so a minimum at (3, 0).

Example 6: tangents and normals

Find the tangent and normal to y = x² at x = 3.

• Point (3, 9); gradient 2x = 6. Tangent: y = 6x − 9.
• Normal gradient −1/6. Normal: y − 9 = −(1/6)(x − 3).

Example 7: parametric differentiation

x = t² and y = 2t. Find dy/dx.

• dy/dx = (dy/dt)/(dx/dt) = 2/(2t) = 1/t.

Example 8: optimisation

Squares of side x cm are cut from each corner of a 12 cm square sheet, which is folded into an open box. Find the maximum volume.

• V = x(12 − 2x)². dV/dx = (12 − 2x)² − 4x(12 − 2x) = (12 − 2x)(12 − 6x).
• dV/dx = 0 gives x = 2 (x = 6 gives no box). V = 2 × 8² = 128 cm³; check it is a maximum with the second derivative or a sign change.

Example 9: connected rates of change

A sphere's radius grows at 0.5 cm/s. How fast is its volume growing when r = 4 cm?

• dV/dt = dV/dr × dr/dt = 4πr² × 0.5 = 4π × 16 × 0.5 = 32π cm³/s.

More mistakes to avoid

• Using degrees with trigonometric derivatives; calculus needs radians.
• Forgetting the chain rule on the inner function, such as sin 3x or e^2x.
• Not justifying the nature of a stationary point.
• Mixing up the gradients of the tangent and the normal.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Which differentiation topics are in Year 1 and Year 2?

Year 1 covers first principles, polynomials, tangents, normals and stationary points. Year 2 adds the chain, product and quotient rules, trigonometric, exponential and log functions, implicit and parametric differentiation, and connected rates of change.

Do I need to learn the quotient rule if I know the product rule?

It is worth knowing both. The quotient rule is quicker for fractions, and some boards include it in the formula booklet; check yours.

Key takeaways

• The derivative is the gradient of the curve at a point.
• Learn the standard results and the chain, product and quotient rules.
• Use radians for trigonometric calculus.
• Classify stationary points with the second derivative or a sign change.
• Show every step; method marks reward clear working.

Where to go next

Differentiation pairs naturally with integration and links to mechanics and optimisation.

• A level Maths integration
• A level Maths mechanics
• A level Maths formula booklet

How to revise differentiation well

Start by writing the standard derivatives from memory on a blank page, then check them against your notes. Next, do ten short questions on each rule in turn: chain, product, quotient. Once each rule feels automatic, switch to mixed questions where you have to decide which rule applies, because that decision is what the exam really tests. Finish with longer problems on stationary points, optimisation and rates of change, where differentiation is only one step in a bigger argument.

When you get a question wrong, write down why. Was it a sign error, a missing chain rule factor, or a misread question? Patterns appear quickly, and fixing your two most common errors often gains more marks than learning anything new.

Getting help

If differentiation still feels shaky, a few targeted sessions on the rules and on exam-style problems usually make a big difference, and a tutor can spot the exact step where your method breaks down.