Differentiation A-Level Maths: Every Rule, Every Exam Application

Why differentiation runs through every A-Level paper

Differentiation isn't just a topic — it's a tool. It appears directly in calculus questions but also implicitly in curve sketching, optimisation, mechanics (velocity and acceleration), and rates of change problems.

A student who can differentiate fluently and accurately can access marks across the entire paper. A student who can't will find large sections of every paper inaccessible.

The core differentiation rules (and when to use each)

Power rule

d/dx(xⁿ) = nxⁿ⁻¹

This works for any constant power — positive, negative, or fractional. Before differentiating, simplify: write √x as x^(1/2), write 1/x² as x⁻².

Chain rule

d/dx(f(g(x))) = f'(g(x)) · g'(x)

Use whenever you have a function of a function: (2x+1)⁵, sin(x²), eˣ³, ln(3x−1). Identify the outer function and inner function. Differentiate the outer (keeping the inner), then multiply by the derivative of the inner.

Product rule

d/dx(uv) = u·dv/dx + v·du/dx

Use whenever two functions are multiplied together: x²sin x, xeˣ, x³ln x.

Memory aid: "first times derivative of second, plus second times derivative of first."

Quotient rule

d/dx(u/v) = (v·du/dx − u·dv/dx) / v²

Use for fractions where both numerator and denominator are functions of x. Note: the minus sign in the numerator means order matters. Numerator is always (bottom × derivative of top) − (top × derivative of bottom).

Implicit differentiation

For equations involving both x and y where you can't easily rearrange for y:

• Differentiate every term with respect to x
• Every time you differentiate a term containing y, multiply by dy/dx (chain rule)
• Collect dy/dx terms and solve

Example: x² + y² = 25

Differentiating: 2x + 2y(dy/dx) = 0

Solving: dy/dx = −x/y

The most tested applications

Finding stationary points

1. 1Differentiate to get dy/dx
2. 2Set dy/dx = 0 and solve for x
3. 3Find y by substituting into original equation
4. 4Determine nature using second derivative: if d²y/dx² > 0, minimum; if < 0, maximum; if = 0, further investigation needed

Optimisation problems

These questions give you a constraint and ask you to minimise or maximise something.

1. 1Express the quantity to minimise/maximise as a function of one variable (use the constraint to eliminate others)
2. 2Differentiate and set to zero
3. 3Verify it's a minimum or maximum using the second derivative
4. 4Check boundary conditions if the domain is restricted

Connected rates of change

Using the chain rule across rates: if you know dV/dt and want dr/dt, and you know V in terms of r:

dV/dt = dV/dr × dr/dt

These questions always give you the related quantities — your job is to connect them through the chain rule.

Parametric differentiation

When x = f(t) and y = g(t):

dy/dx = (dy/dt) ÷ (dx/dt)

For the second derivative: d²y/dx² = (d/dt(dy/dx)) ÷ (dx/dt)

The errors that cost A-level students marks

1. 1Forgetting to apply the chain rule — d/dx(sin(2x)) = cos(2x), not cos(2x). The inner derivative (2) must multiply.

1. 1Product rule sign errors — Always write u, v, du/dx, dv/dx separately before applying the formula

1. 1Implicit differentiation — forgetting the dy/dx when differentiating terms containing y: d/dx(y²) = 2y(dy/dx), not just 2y

1. 1Not checking stationary point nature — A question asking for a maximum or minimum requires confirmation via second derivative or sign change. A stationary point isn't automatically a maximum or minimum.

1. 1Rates of change sign — If a quantity is decreasing, its rate of change is negative. Don't lose the sign.

How to build differentiation fluency

Generate 20 differentiation questions in Infinity Stars mixing: basic power rule, chain rule, product rule, implicit, and connected rates. Don't label what technique you need — that's the examiner's job, not yours.

In the real exam, the technique isn't named. "Find dy/dx" for x²sin x is a product rule question. "Find the gradient of the curve x² + 3xy + y² = 7 at the point (1,1)" is implicit differentiation. You need to identify and execute without prompting.

That identification skill only develops through unseen practice.