A-Level Maths: Differentiation Techniques

Differentiation technique is one of the most frequently used skills in A-Level Mathematics: the product, quotient and chain rules and their combinations appear across pure, mechanics and modelling questions alike. This quiz drills the complete technical repertoire.

It opens where the specification does, with first principles: the limit definition applied to x², the derivation every board reserves the right to examine. The special derivatives follow — a question on which functions equal their own derivative (eˣ and its constant multiples, the property that defines the exponential), the derivatives of sin and cos as a paired recognition, and ln(5x), where the chain rule's factor of 5 cancels to leave 1/x, a result that surprises students the first time.

The composite rules get one question each in their cleanest exam forms: the product rule on x²sin x, the chain rule on sin(x³) — with distractors built from the two classic errors, differentiating only the outside or only the inside — and a rule-selection question asking which expressions genuinely require the chain rule.

The Year 13 techniques complete the set: implicit differentiation on the circle x² + y² = 25, producing dy/dx = −x/y through the y-term's chain-rule factor; parametric differentiation on x = t², y = t³ via the quotient of derivatives; and a second-derivative computation on e^(2x), where each differentiation multiplies by 2 to give 4e^(2x).

Each derivative in the set has been checked. Explanations show the rule invocation step by step, in the layout the written paper's method marks reward. Pair it with the Applications of Differentiation quiz for the complete calculus picture.

• Differentiate from first principles using the limit definition
• Apply product, quotient and chain rules, alone and in combination
• Recall the derivatives of sin, cos, e^x and ln x
• Differentiate implicitly and handle the dy/dx terms correctly
• Differentiate parametric curves via dy/dt over dx/dt

Topic scope follows section G (Differentiation) of the DfE's prescribed AS and A level mathematics subject content — 100% common across all exam boards: first principles, the product, quotient and chain rules, standard derivatives, and implicit and parametric differentiation.