Basic Rules:

• d/dx($x^n$) = nx^(n-1), d/dx(c) = 0
• (fg)' = f'g + fg', $\frac{f}{g}$' = $\frac{f'g-fg'}{g}$^2
• d/dx[f(g(x))] = f'(g(x))*g'(x)

Trigonometric: sin->cos, cos->-sin, tan->$sec^2$, cot->-$cosec^2$, sec->sectan, cosec->-coseccot

Inverse Trig: sin^(-1)->1/sqrt(1-$x^2$), cos^(-1)->-1/sqrt(1-$x^2$), tan^(-1)->$\frac{1}{1+x^2}$, cot^(-1)->-$\frac{1}{1+x^2}$

Exponential/Log: $e^{x-}$>$e^x$, $a^{x-}$>$a^x$*ln a, ln x->1/x, $log_a$ x->$\frac{1}{x*ln a}$

Key Simplifications (substitute x = tan t):

• tan^(-1)(2$\frac{x}{1-x^2}$) = 2tan^(-1)(x)
• sin^(-1)(2$\frac{x}{1+x^2}$) = 2tan^(-1)(x)
• cos^(-1)$\frac{(1-x^2}{(1+x^2)}$) = 2tan^(-1)(x)

Key Simplifications (substitute x = sin t):

• sin^(-1)(2x*sqrt(1-$x^2$)) = 2sin^(-1)(x)
• sin^(-1)(3x-4$x^3$) = 3sin^(-1)(x)

Parametric: dy/dx = $\frac{dy/dt}{(dx/dt)}$, $d^{2y}$/$dx^2$ = [d/dt$\frac{dy}{dx}$]/$\frac{dx}{dt}$

Logarithmic: For y = $f^g$: dy/dx = $f^g$ * [g'ln f + gf'/f]

Leibniz: (uv)^(n) = sum C(n,r)*u^(n-r)*v^(r)

nth derivatives: ($e^{ax}$)^(n) = $a^n$*e^(ax), (sin(ax+b))^(n) = $a^n$sin(ax+b+npi/2)