Pure Mathematics
Differentiation
Power rule$$\frac{d}{dx}(x^n) = nx^{n-1}$$
Product$$\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$$
Quotient$$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v u' - u v'}{v^2}$$
Chain$$\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$$
Integration
$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad n \ne -1$$
$$\int \frac{1}{x} \, dx = \ln|x| + C$$
By parts$$\int u \, dv = uv - \int v \, du$$
Trigonometric identities
$$\sin^2\theta + \cos^2\theta = 1$$
$$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$$
$$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$$
Binomial expansion
$$(a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$$
📚 More topics to be added: sequences & series, exponentials/logs, vectors, coordinate geometry, parametrics. This scaffold is ready to accept contributions.
Statistics
Binomial distribution
$$X \sim B(n, p) \Rightarrow P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}$$
Normal distribution
$$Z = \frac{X - \mu}{\sigma}, \quad Z \sim N(0, 1)$$
Test statistic for correlation ($r$)
$$t = r\sqrt{\frac{n-2}{1-r^2}}$$
📚 To be added: Poisson (IAL/CIE only), regression lines, hypothesis test frameworks.
Mechanics
Kinematics (constant acceleration)
$$v = u + at, \quad s = ut + \tfrac{1}{2}at^2, \quad v^2 = u^2 + 2as$$
Newton's 2nd law
$$F = ma$$
Energy
$$\text{KE} = \tfrac{1}{2}mv^2, \quad \text{PE} = mgh$$
📚 To be added: moments, projectile trajectory formulas, impulse & momentum, circular motion.