✕ Calculus

Calculus, quick reference

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Rules, standard derivatives and integrals, and the complex-number toolkit, condensed for revision.

Complex numbers (91577)

Basics i² = −1 ; z = a+bi ; z̄ = a−bi ; |z| = √(a²+b²) Polar z=r\,\mathrm{cis}\,\theta\qquad r=|z|\qquad \tan\theta=\frac{b}{a} Multiply / divide moduli ×÷ ; arguments +− De Moivre (r cis θ)ⁿ = rⁿ cis(nθ) nth roots z^{1/n}=r^{1/n}\,\mathrm{cis}\!\left(\frac{\theta+2\pi k}{n}\right),\ k=0,\dots,n-1

Differentiation rules (91578)

Power \frac{d}{dx}x^n=nx^{n-1} Chain d/dx f(g(x)) = f′(g) · g′ Product (uv)′ = u′v + uv′ Quotient \left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime}v-uv^{\prime}}{v^2}

Standard derivatives

Trig sin→cos ; cos→−sin ; tan→sec²x Exp / log \frac{d}{dx}e^{kx}=ke^{kx}\qquad \frac{d}{dx}\ln x=\frac{1}{x}\qquad \frac{d}{dx}\ln f=\frac{f^{\prime}}{f} Turning points y′ = 0 ; y″<0 max, y″>0 min

Integration (91579)

Power \int x^n\,dx=\frac{x^{n+1}}{n+1}+c\quad(n\neq-1) Reciprocal / exp \int\frac{1}{x}\,dx=\ln|x|+c\qquad \int e^{kx}dx=\frac{1}{k}e^{kx}+c Trig ∫ sin = −cos ; ∫ cos = sin ; ∫ sec² = tan Area A = ∫ₐᵇ y dx ; between: ∫(y_top − y_bottom) dx Volume of revolution V = π ∫ₐᵇ y² dx (about x-axis) Separable DE separate variables, integrate both sides, +c