AQA A-Level Mathematics: Differentiation (7357)

This guide covers Section G: Differentiation, G1 through G6, of AQA A-Level Mathematics (7357), first teaching September 2017. Like the rest of the Pure content, Section G is Department for Education (DfE) prescribed content shared across every exam board offering A-level Mathematics, not set independently by AQA. Section G runs across both Paper 1 (pure) and Paper 2 (pure and mechanics/statistics), so fluency here underpins marks across the whole qualification, not just one exam paper.

Where this fits in 7357

Differentiation builds directly on Section B (Algebra and Functions) and Section D (Coordinate Geometry), applying the algebraic and graph-sketching skills developed there to the new question of how a function’s rate of change behaves. It also underpins Section H (Integration), which reverses the operation, and appears throughout the mechanics content (rates of change of displacement and velocity) and statistics content examined on Paper 2 and Paper 3.

Syllabus coverage

AQA A-LEVEL MATHEMATICS (7357) — SECTION G: DIFFERENTIATION

• G1 — understand and use the derivative as the gradient of the tangent to a curve, understand the gradient function and the second derivative, and understand differentiation from first principles for small positive integer powers of x and for sin x and cos x
• G2 — differentiate xⁿ, eᵏˣ, aᵏˣ, sin kx, cos kx, tan kx and ln x, and their sums and constant multiples
• G3 — apply differentiation to find gradients, tangents and normals, maxima and minima, stationary points, points of inflection, and where a function is increasing or decreasing
• G4 — differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions
• G5 — differentiate simple functions defined implicitly or parametrically, for first derivatives only
• G6 — construct simple differential equations in pure mathematics and in a variety of contexts, including kinematics, population growth, and the relationship between price and demand

G1: what a derivative actually is

The derivative of f(x) at a point is the gradient of the tangent to y = f(x) at that point, formally defined as the limit of the gradient of a chord as the two points on the curve move together:

f'(x) = lim(h->0) [f(x+h) - f(x)] / h

Keep two interpretations of this same quantity distinct: geometrically it is the gradient of the tangent line; physically it is the instantaneous rate of change of the quantity the function represents. You are expected to sketch a gradient function from a given curve – where the original function is increasing, its gradient function is positive; where the original has a turning point, its gradient function crosses zero – and to use the second derivative as the rate of change of the gradient itself, which connects directly to identifying convex or concave sections of a curve and points of inflection. First-principles differentiation is examinable specifically for small positive integer powers of x and for sin x and cos x, not for arbitrary functions.

G2: standard derivatives

Function	Derivative
xⁿ (n rational)	nxⁿ⁻¹
eᵏˣ	k eᵏˣ
aᵏˣ	k aᵏˣ ln a
sin kx	k cos kx
cos kx	-k sin kx
tan kx	k sec²kx
ln x	1/x

These extend to constant multiples, sums and differences by linearity – differentiate term by term. The aᵏˣ result is the one most often mis-recalled under exam pressure, since it needs an extra ln a factor that eᵏˣ does not, arising because aᵏˣ can itself be rewritten as eᵏˣ ln a before the chain rule is applied.

G3: gradients, tangents and stationary points

Beyond finding a gradient at a point (and the corresponding normal gradient, which is -1 divided by the tangent gradient), G3 covers finding and classifying stationary points. Setting f’(x) = 0 and solving locates candidate stationary points; classifying them uses the second derivative – f’‘(x) > 0 indicates a minimum, f’‘(x) < 0 indicates a maximum, and f’‘(x) = 0 requires further investigation, typically by checking the sign of f’(x) on either side of the point. A point of inflection occurs where f’‘(x) = 0 and the concavity of the curve genuinely changes either side of that point, which is a distinct condition from f’‘(x) = 0 alone. G3 also covers determining where a function is increasing (f’(x) > 0) or decreasing (f’(x) < 0) across an interval.

Worked example: classifying a stationary point

Find and classify the stationary points of f(x) = x³ - 3x² - 9x + 5.

f'(x) = 3x² - 6x - 9 = 3(x² - 2x - 3) = 3(x - 3)(x + 1)
f'(x) = 0  =>  x = 3 or x = -1

f''(x) = 6x - 6
f''(3)  = 18 - 6 = 12 > 0   =>  minimum at x = 3
f''(-1) = -6 - 6 = -12 < 0  =>  maximum at x = -1

Stopping at f’(x) = 0 without applying the second-derivative test is the single most common way marks are lost on this sub-topic when a question explicitly asks for the nature of each stationary point.

G4: product, quotient and chain rules

Rule	Formula
Product	(uv)’ = u’v + uv’
Quotient	(u/v)’ = (u’v - uv’) / v²
Chain	dy/dx = dy/du x du/dx

These extend to connected rates of change (using the chain rule to relate, for example, dV/dt to dV/dr x dr/dt) and to differentiating inverse functions. Connected-rates-of-change problems are a classic source of lost marks purely through not identifying which two rates the chain rule needs to link – write out explicitly which variable each given rate is with respect to before attempting the chain-rule step, rather than attempting the algebra before the relationship between variables is clear.

G5: implicit and parametric differentiation

For an implicitly-defined relation such as x² + y² = r², differentiate both sides term by term with respect to x, treating y as a function of x – so d(y²)/dx becomes 2y dy/dx via the chain rule – then rearrange for dy/dx. For a parametrically-defined curve, with x = f(t) and y = g(t), find dy/dx via dy/dx = (dy/dt) divided by (dx/dt). Only the first derivative is required for implicit and parametric relations at this level; second derivatives for these two specific techniques are not part of the specification’s requirements here.

G6: forming differential equations

The specification expects candidates to construct simple differential equations, both in pure mathematics contexts and in named modelling contexts: kinematics, population growth, and the relationship between price and demand. The skill genuinely being tested is translating a verbal rate-of-change statement into a correct equation – for example, “the rate of increase of a population is proportional to the population’s current size” becomes dP/dt = kP – not necessarily solving the resulting equation, since differential-equation solving methods belong to Section H, Integration, rather than Section G.

How to approach it

Because Section G spans first principles, a table of standard derivatives, several differentiation rules, and two distinct non-standard forms (implicit and parametric), the most efficient revision structure separates “recall” content (the standard derivatives table, the rule formulas) from “apply” content (classifying stationary points, connected rates of change, constructing differential equations), since exam questions test both in combination but they are learned differently – one through repetition until automatic, the other through worked practice on varied scenarios. Because G6 deliberately stops short of solving the equations it asks you to construct, resist the instinct to solve a constructed differential equation using G-section techniques; recognise where G6’s task actually ends.

Official syllabus

AQA A-Level Mathematics (7357) specification, Section G Differentiation, for first teaching from September 2017 – aqa.org.uk, content shared with the Department for Education’s prescribed A-level Mathematics content and content shared across exam boards. Verified 2026-09-05.