Revision sheet · Mathematics

Limits and derivatives

The limit of (xⁿ − aⁿ) ÷ (x − a): $lim_{x \to a} \frac{x ^{n} - a ^{n}}{x - a} = n a^{n - 1}$
The limit of sin x ÷ x: $lim_{x \to 0} \frac{s i n x}{x} = 1$
Derivative of xⁿ (the power rule): $\frac{d}{d x} x^{n} = n x^{n - 1}$
The product rule: $\frac{d}{d x} (uv) = u \frac{d v}{d x} + v \frac{d u}{d x}$
The quotient rule: $\frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v ^{2}}$
Derivative of sin x: $\frac{d}{d x} sin x = cos x$
Derivative of cos x: $\frac{d}{d x} cos x = - sin x$