A-Level · A-Level Maths
A-Level Maths chain rule
What is chain rule?
The chain rule differentiates a composite function: if y = f(g(x)) then dy/dx = f'(g(x)) · g'(x). A-Level Maths students apply it across every paper — pure (e.g. sin(2x), (1+x²)⁵), mechanics (variable acceleration) and statistics (likelihood functions). The trick is to spot the inner function first, then differentiate from the outside in.
Worked example
Differentiate y = (3x² + 1)⁴.
- Identify the inner function: u = 3x² + 1, so y = u⁴.
- Differentiate the outer function with respect to u: dy/du = 4u³.
- Differentiate the inner function with respect to x: du/dx = 6x.
- Apply the chain rule: dy/dx = dy/du · du/dx = 4u³ · 6x = 24x(3x² + 1)³.
When to use the chain rule
Whenever the function you're differentiating is a function inside another function. Common signals:
- Bracketed expressions raised to a power: (anything)ⁿ.
- Trig functions of non-x arguments: sin(2x), cos(3x²), tan(πx).
- Exponential or logarithmic functions of expressions: e^(2x+1), ln(x² + 4).
- Composite functions appearing inside the product or quotient rule.
Where it appears in the A-Level Maths exam
AQA 7357, Edexcel 9MA0 and OCR H240 examine the chain rule on Pure papers (typically Paper 1 or Paper 2) and synoptically inside questions on related rates, implicit differentiation, parametric equations and integration by substitution.
Common mistakes
- Forgetting to multiply by the derivative of the inner function (the most common error).
- Differentiating the outer function with respect to x instead of u.
- Mishandling negative powers — d/dx[(2x+1)^(-2)] = -2(2x+1)^(-3) · 2, not just -2(2x+1)^(-3).
- Applying the chain rule once instead of recursively for triple composites.
Frequently asked
- When do I use the chain rule vs the product rule?
- Chain rule: composite functions (a function inside another function). Product rule: two functions multiplied together. Often you need both — for example, d/dx[x² · sin(3x)] requires the product rule for the outer multiplication and the chain rule inside the derivative of sin(3x).
- Is the chain rule on the formula booklet?
- No. The chain rule is examinable without a formula prompt on AQA, Edexcel and OCR. Memorise the form dy/dx = dy/du · du/dx.
- How do I use the chain rule on parametric equations?
- For x = x(t) and y = y(t), the chain rule gives dy/dx = (dy/dt) ÷ (dx/dt). Differentiate each parametric component with respect to t, then divide.
A-Level Maths glossary terms
- Chain ruleThe chain rule differentiates composite functions: if y = f(g(x)) then dy/dx = f'(g(x)) · g'(x). It appears across A-Level Maths and Further Maths whenever an outer function wraps an inner function — common examples are sin(2x), e^(3x²) and (1+x²)⁵. Without the chain rule, every composite differentiation drops marks; with it, the standard trick is to spot the inner function first.
- Implicit differentiationImplicit differentiation differentiates equations where y is not isolated, e.g. x² + y² = 25. Treat y as a function of x and apply the chain rule whenever you differentiate a y term — d(y²)/dx becomes 2y · dy/dx. Used heavily for circles, ellipses and tangent/normal problems where rearranging to y = f(x) is impractical.
- Differential equationsA differential equation relates a function to its derivatives. A-Level Maths examines first-order separable equations (dy/dx = f(x)·g(y)), solved by separating variables and integrating both sides. Applications include exponential growth/decay (dN/dt = kN), Newton's law of cooling and rate-of-reaction models. Further Maths extends to second-order equations and integrating factors.
- Parametric equationsParametric equations express x and y separately as functions of a parameter t, e.g. x = cos(t), y = sin(t). They model curves that aren't conveniently expressed as y = f(x) — including circles, ellipses, and projectile paths. A-Level Maths uses the chain rule dy/dx = (dy/dt) / (dx/dt) to find gradients, and substitution to eliminate the parameter and recover the Cartesian equation.
Related on StudyVector
Last updated: . StudyVector is independent and is not affiliated with AQA, Edexcel, OCR or JCQ. For canonical specification wording, see the awarding body's published documents.