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A Levels / Maths

Differentiation

Learn A-Level differentiation from first principles, implicit and parametric differentiation, chain, product and quotient rules, second derivatives, trigonometric differentiation and applications of derivatives.

Differentiation measures the instantaneous rate of change of a function and gives the gradient of the tangent to a curve at a particular point.
Differentiation from first principles uses the limit definition f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h} and can be used to derive derivatives of small positive integer powers of xx.
The first principles formula explains why differentiation gives a gradient by considering the limiting gradient of a secant line as two points on a curve move together.
A second derivative, written d2ydx2\frac{d^2y}{dx^2}, describes how the gradient itself changes and can be used to study concavity, convexity and points of inflection.
Stationary points occur where dydx=0\frac{dy}{dx}=0, and the sign of the first or second derivative can be used to classify local maxima and minima.
Equations of tangents use the gradient dydx\frac{dy}{dx} at a point, while the gradient of a normal is the negative reciprocal of the tangent gradient when the tangent is not horizontal or vertical.
The product rule is used when differentiating a product of two functions and is given by ddx(uv)=udvdx+vdudx\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}.
The quotient rule differentiates a quotient of two functions using ddx(uv)=vdudx−udvdxv2\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}.
The chain rule is used for composite functions and can be written as dydx=dydududx\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}, making differentiation using chain rule essential for expressions containing functions within functions.
Implicit differentiation is used when xx and yy are related by an equation that is not written explicitly as y=f(x)y=f(x); terms containing yy must be differentiated using the chain rule.
Parametric equations define xx and yy in terms of a parameter, and parametric differentiation uses dydx=dy/dtdx/dt\frac{dy}{dx}=\frac{dy/dt}{dx/dt} when dxdt≠0\frac{dx}{dt}\ne0.
The derivative of ln⁡x\ln x is 1x\frac{1}{x} for x>0x>0, while the derivative of exe^x is exe^x, making exponential and logarithmic functions especially important in calculus.
Trig differentiation includes ddx(sin⁡x)=cos⁡x\frac{d}{dx}(\sin x)=\cos x, ddx(cos⁡x)=−sin⁡x\frac{d}{dx}(\cos x)=-\sin x and the derivative of tanx, ddx(tan⁡x)=sec⁡2x\frac{d}{dx}(\tan x)=\sec^2x.
Rational and fractional powers can be differentiated using the power rule ddx(xn)=nxn−1\frac{d}{dx}(x^n)=nx^{n-1}, which also gives the derivative of 1/x1/x by writing it as x−1x^{-1}.
Differentiation can be applied to increasing and decreasing functions, maxima and minima, connected rates of change, inverse functions, kinematics, simple differential equations and mathematical models of growth.
A Level Differentiation | Chain, Product & Implicit Rules