A Level Maths: Further Differentiation Topic Summary and Resources
Video Lessons
Watch alongside the worksheet for the full lesson experience, then test your understanding with the lesson questions.
Revision Notes
Handwritten notes summarising the key ideas for each lesson. Ideal for quick review before a test.
Exam Questions
Past-paper-style questions organised by topic, with full mark schemes.
- Differentiation (Chain, Product and Quotient Rules) Introductory (OCR)
- Differentiation (Chain, Product and Quotient Rules) Introductory (Edexcel)
- Differentiation (Chain, Product and Quotient Rules) Harder (OCR)
- Implicit Differentiation (OCR)
- Implicit Differentiation (Edexcel)
- Connected Rates of Change (OCR)
- Connected Rates of Change (Edexcel)
Drawn from OCR and Edexcel past papers but designed to be useful for students of all UK exam boards — including AQA and OCR MEI — unless a sheet is explicitly board-specific.
Before You Start This Topic
It will help if you are confident with the following:
- Differentiation — Year 1 differentiation is the immediate prerequisite
- Trigonometry — Compound and Double Angle Formulae}}: needed for many trig differentiation problems
- Exponentials and Logarithms — needed for differentiating $e^x$ and $\ln(x)$
- Functions — composite and inverse function concepts are essential
A Level Maths further differentiation is the Year 2 calculus topic that extends Differentiation from simple powers of $x$ to a much broader set of functions and techniques. You learn the chain, product, and quotient rules; how to differentiate trigonometric, exponential, and logarithmic functions; and how to handle inverse and implicit functions. Mastery here unlocks Connected Rates of Change, Parametric Equations, Further Integration, and Differential Equations.
You meet the three workhorse rules: the chain rule for composite functions, the product rule for differentiating $uv$, and the quotient rule for differentiating $u/v$. You differentiate trig functions including $\sin$, $\cos$, $\tan$, $\sec$, $\csc$, and $\cot$, and the inverse trig functions $\arcsin$, $\arccos$, $\arctan$. You differentiate $e^x$, $a^x$, $\ln(x)$, and combinations of these with other functions. Implicit differentiation lets you find $\frac{dy}{dx}$ when $y$ is not explicitly written as a function of $x$ — useful for circles, ellipses, and more complex curves. You also differentiate inverse functions using the relationship $\frac{dy}{dx} = \frac{1}{dx/dy}$.
Further differentiation is part of the Pure Maths strand of A Level Maths for AQA, Edexcel, OCR, and OCR MEI students.
Watch out for…
A few things to be careful with: the chain rule is required whenever you differentiate a composite function, so $f'(g(x))$ needs multiplying by $g'(x)$; the product rule is $u \frac{dv}{dx} + v \frac{du}{dx}$, NOT $\frac{du}{dx} \cdot \frac{dv}{dx}$; for implicit differentiation, every time $y$ is differentiated you must multiply by $\frac{dy}{dx}$ (chain rule); and a common slip with trig differentiation is forgetting the chain rule for $\cos(2x)$, which differentiates to $-2\sin(2x)$, not $-\sin(2x)$.