Basic Rules:

• d/dx(x^n) = nx^(n-1), d/dx(c) = 0
• (fg)' = f'g + fg', (f/g)' = (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)->1/(1+x^2), cot^(-1)->-1/(1+x^2)

Exponential/Log: e^x->e^x, a^x->a^xln a, ln x->1/x, log_a x->1/(xln a)

Key Simplifications (substitute x = tan t):

• tan^(-1)(2x/(1-x^2)) = 2tan^(-1)(x)
• sin^(-1)(2x/(1+x^2)) = 2tan^(-1)(x)
• cos^(-1)((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-4x^3) = 3sin^(-1)(x)

Parametric: dy/dx = (dy/dt)/(dx/dt), d^2y/dx^2 = [d/dt(dy/dx)]/(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^ne^(ax), (sin(ax+b))^(n) = a^nsin(ax+b+n*pi/2)