Pure Mathematics 4 (A Level) – Formula List

1. Parametric Differentiation

First Derivative

$\frac{dy}{dx}=\frac{dy}{dt}÷\frac{dx}{dt}$

Used when both x and y are functions of a parameter t.

Second Derivative

$\frac{d^{2}y}{d{x}^2}=\frac{d}{dt}\left(\frac{dy}{dx}\right)÷\frac{dx}{dt}$

Required for curvature, nature of stationary points, and concavity.

2. Polar Coordinates

Cartesian Conversion

$x=r\cos \theta$
$y=r\sin \theta$

Converts polar equations into Cartesian form.

Area in Polar Coordinates

$A=\frac{1}{2}{\int }_{\alpha }^{\beta }{r}^{2}d\theta$

Used to find the area enclosed by a polar curve.

3. Vector Algebra

Vector Equation of a Line

$r=a+\lambda b$

a is a point vector, b is a direction vector.

Dot Product

$a·b=\mathrm{|a|}\mathrm{|b|}\cos \theta$

Used to find angles and check perpendicularity.

Angle Between Two Lines

$\cos \theta =\frac{a·b}{\mathrm{|a|}\mathrm{|b|}}$

4. Differential Equations

Separable Form

$\frac{\mathrm{dy}}{\mathrm{dx}}=f\left(x\right)g\left(y\right)$

Rearrange and integrate both sides.

General Solution

${\int }_{}^{}g\left(y\right)dy={\int }_{}^{}f\left(x\right)dx+C$

5. Numerical Methods

Newton–Raphson Formula

$x_{n+1}=x_n-\frac{f\left(x_n\right)}{f\text{'}\left(x_n\right)}$

Used to find roots of equations iteratively.

6. Integration Techniques

Integration by Parts

${\int }_{}^{}udv=uv-{\int }_{}^{}vdu$

Reduction Formula (General Form)

$I_n=f\left(I_{n-1}\right)$

Used for powers of trig or exponential integrals.