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LearnCambridge97092026–2027Paper 3 · Pure Mathematics 3Differentiation

Paper 3 · Module

Differentiation

5 subtopics≈ 215 min in total

Start the module: Derivatives of eˣ, ln x and trigonometric functions Open graphing calculator

Derivatives of eˣ, ln x and trigonometric functions
The derivatives of $e^x$ , $\ln x$ , $\sin x$ , $\cos x$ and $\tan x$ , why they need radians, and how to combine them with the chain rule for functions like $e^{3x}$ , $\ln(x^2+1)$ and $\sin^2x$ .
≈ 40 min
The product and quotient rules
How to differentiate two functions multiplied together, $uv$ , or divided, $\dfrac uv$ , and how to tidy the result by factorising, so it is ready for finding stationary points.
≈ 45 min
Implicit differentiation
Differentiating curves like $x^2+xy+y^2=12$ that can't easily be written as $y=\ldots$ : differentiate every term with respect to $x$ , remembering that $y$ terms pick up a factor $\dfrac{dy}{dx}$ .
≈ 45 min
Parametric differentiation
Curves described by a third variable $t$ : $x=f(t)$ , $y=g(t)$ . How to find their gradient with $\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$ , and how to find tangents and special points.
≈ 40 min
Stationary points and tangents
Putting the Pure 2/3 rules to work: finding exact stationary points of curves like $y=xe^{2x}$ and $y=\dfrac{\ln x}{x}$ , deciding their nature, and finding tangents and normals.
≈ 45 min