Calculus
Differentiation
From first principals
f'(x) = lim f(x+h)-f(x)
...............…..h
Stationary points
f'(x) = 0
Chain rule
dy/dx = dy/du x du/dx
Points of inflection
f''(x) = 0
e^x, ln x, a^x
d/dx e^f(x) = f'(x)e^f(x)
d/dx ln f(x) = f'(x)/f(x)
d/dx a^x = a^x ln a
Trig
d/dx sin ax = acos x
d/dx cos ax = -asin x
d/dx tan ax = asec^2 x
Product rule
y = f(x)g(x)
dy/dx = f'(x)g(x) + f(x)g'(x)
d/dx cosec x = -cosec x cot x
d/dx sec x = sec x tan x
d/dx cot x = -cosec^2 x
Quotient rule
y = f(x)/g(x)
dy/dx = f'(x)g(x) - f(x)g'(x)
.....................g(x)^2
Integration
e^x, 1/x
∫ e^f(x) dx = 1/f'(x)e^f(x) + C
∫ 1/f(x) dx = 1/f'(x) ln|f(x)| + C
∫ f'(x)/f(x) dx = ln|f(x)| + C
Trig
∫ sin x dx = -cos x + C
∫ cos x dx = sin x +C
∫ sec^2 x dx = tan x + C
Substitution
∫ f'(x)[f(x)]^n dx
u = f(x)
dx = du/f(x)
Parts
∫ u dv/dx dx = uv - ∫ v du/dx dx
∫ tan x dx = ln|sec x| + C
∫ cot x dx = ln|sin x| + C