A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Differentiation of Implicit and Parametric Functions — mark scheme explained
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
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For implicit differentiation, marks are typically awarded for correctly differentiating both sides of the equation, collecting terms involving dy/dx, and solving for dy/dx. For parametric differentiation, marks are given for finding dx/dt and dy/dt, using the correct formula to find dy/dx, and simplifying the final answer.
What the command words demand
- Differentiate
- Find the derivative of the given function or relation.
- Find dy/dx
- Determine the first derivative with respect to x for the given implicit or parametric function.
- Simplify
- Reduce the expression to its simplest form.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 5-7 minutes per question for implicit or parametric differentiation problems in the exam.
- Differentiate both sides of the equation with respect to x: 2x + 2y(dy/dx) = 0 .1 mark
- Collect all terms involving dy/dx on one side: 2y(dy/dx) = -2x .1 mark
- Solve for dy/dx: dy/dx = -x/y .1 mark
Final answer: dy/dx = -x/y
Work through every step correctly and you earn all 3 marks.
Another worked example
Given the parametric equations x = t 2 and y = t 3 , find dy/dx.
- Find dx/dt: dx/dt = 2t .1 mark
- Find dy/dt: dy/dt = 3t 2 .1 mark
- Use the formula: dy/dx = (dy/dt) / (dx/dt) = (3t 2 ) / (2t) = 3t/2 .2 marks
Final answer: dy/dx = 3t/2
Work through every step correctly and you earn all 4 marks.
Common mistakes
Forgetting to use the chain rule when differentiating terms involving y in implicit functions.
Why it happens: Students often treat y as a constant instead of a function of x, leading to incorrect differentiation.
Fix: Always remember that y is a function of x and apply the chain rule accordingly.
Making algebraic errors when simplifying expressions after implicit differentiation.
Why it happens: Algebra can be complex, and students may make simple mistakes like sign errors or incorrect factoring.
Fix: 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.
Why it happens: Students may skip this crucial step, making it difficult to solve for dy/dx.
Fix: Always collect and isolate dy/dx before solving for it.
Using the wrong formula for parametric differentiation.
Why it happens: Students may confuse the formula for dy/dx with other calculus formulas.
Fix: 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.
Why it happens: Students may rush through problems and not verify their solutions.
Fix: 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.
Why it happens: Students may leave their answers in a more complex form than necessary.
Fix: Simplify your final answer as much as possible to make it clear and concise.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Implicit Differentiation | Differentiate an implicit equation and rearrange to find dy/dx in terms of x and y. | 3 |
| Parametric Differentiation | Differentiate parametric equations and combine them to find dy/dx in terms of t. | 4 |
| Implicit Differentiation | Differentiate an implicit equation and rearrange to find dy/dx in terms of x and y. | 5 |
| Parametric Differentiation | Find dy/dx from parametric equations by differentiating each with respect to t. | 4 |
| Total across these question types | 16 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.