A-Level · Mathematics · AQA · Mark scheme decoded

AQA A-Level Mathematics: Differentiation of Implicit and Parametric Functions — mark scheme explained

Machine-verifiedchecked against the AQA A-Level Mathematics specificationlast verified 2 July 2026

The short answer

Differentiation is a fundamental concept in calculus, allowing us to find the rate at which one quantity changes with respect to another. In AQA A-Level Mathematics, you will encounter functions that are defined implicitly or parametrically. This section focuses on differentiating these types of functions and relations, specifically for finding the first derivative.

The question

Differentiate the implicit function x 2 + y 2 = 1 with respect to x. [Paraphrased for study — not reproduced from any exam paper.]

Mark scheme, decoded

What each mark is really for — in plain English — and the wording trap that loses it.

S1
Differentiate both sides of the equation with respect to x: 2x + 2y(dy/dx) = 0 .
S2
Collect all terms involving dy/dx on one side: 2y(dy/dx) = -2x .
S3
Solve for dy/dx: dy/dx = -x/y .

Model answer

Worked through, with each step tagged to the mark it earns.

S1
Differentiate both sides of the equation with respect to x: 2x + 2y(dy/dx) = 0 .
S2
Collect all terms involving dy/dx on one side: 2y(dy/dx) = -2x .
S3
Solve for dy/dx: dy/dx = -x/y .
Final answer: dy/dx = -x/y

Common mistakes

Forgetting to use the chain rule when differentiating terms involving y in implicit functions. — Always remember that y is a function of x and apply the chain rule accordingly.
Making algebraic errors when simplifying expressions after implicit differentiation. — Double-check your algebra at each step to ensure accuracy.
Forgetting to collect all terms involving dy/dx on one side of the equation in implicit differentiation. — Always collect and isolate dy/dx before solving for it.
Using the wrong formula for parametric differentiation. — Memorize and understand the correct formula: dy/dx = (dy/dt) / (dx/dt).
Failing to check the consistency of the final answer by substituting values back into the original equations. — Always substitute values back into the original equations to ensure your solution is correct.
Not simplifying the final answer for dy/dx in parametric differentiation. — Simplify your final answer as much as possible to make it clear and concise.

Where the marks go

Full worked solution (all marking points)3 marks