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LearnAQA7357First teaching 2017Paper 1 · PureDifferentiation

Paper 1 · Module

Differentiation

10 subtopics≈ 435 min in total

Start the module: Gradients and the power rule Open graphing calculator

Gradients and the power rule
What the gradient of a curve means, how chords close in on the tangent, and the power rule $\dfrac{d}{dx}x^n=nx^{n-1}$ that makes differentiating fast, including roots, fractions and second derivatives.
≈ 45 min
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
The chain rule
How to differentiate a function of a function, such as $(2x-5)^4$ or $\sqrt{x^2+5}$ , without multiplying out: differentiate the outside, keep the inside, and multiply by the derivative of the inside.
≈ 40 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
Tangents and normals
Using $\dfrac{dy}{dx}$ to find the equation of the tangent and the normal to a curve at a point, points where the tangent has a given gradient, and the follow-on problems examiners love.
≈ 40 min
Stationary points and increasing functions
Where a curve turns: finding stationary points with $\dfrac{dy}{dx}=0$ , deciding whether each is a maximum or minimum with the second derivative, and finding where a function is increasing or decreasing.
≈ 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
Optimisation and rates of change
Using derivatives to answer real questions: the largest area, the smallest surface, the cheapest design, and how fast one quantity changes when another one does.
≈ 50 min