Differentiation

Standard derivatives

$\frac{d}{d x} x^{n} = n x^{n - 1},$ $\frac{d}{d x} e^{x} = e^{x},$ $\frac{d}{d x} a^{x} = a^{x} ln a,$ $\frac{d}{d x} ln x = \frac{1}{x}$
$(sin x)^{'} = cos x, (cos x)^{'} = - sin x, (tan x)^{'} = sec^{2} x, (sec x)^{'} = sec x tan x, (csc x)^{'} = - csc x cot x, (cot x)^{'} = - csc^{2} x$
$(sin^{- 1} x)^{'} = \frac{1}{1 - x ^{2}},$ $(cos^{- 1} x)^{'} = - \frac{1}{1 - x ^{2}},$ $(tan^{- 1} x)^{'} = \frac{1}{1 + x ^{2}},$ $(sec^{- 1} x)^{'} = \frac{1}{∣ x ∣ x ^{2} - 1}$

$(uv)^{'} = u^{'} v + u v^{'},$ $(\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}},$ $\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$
Parametric
$\frac{d y}{d x} = \frac{d y / d t}{d x / d t},$ $\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d t} (\frac{d y}{d x}) / \frac{d x}{d t}$
Logarithmic
$y = u^{v} \Rightarrow \frac{y ^{'}}{y} = v^{'} ln u + \frac{v u ^{'}}{u}$
Implicit
$F (x, y) = 0 \Rightarrow \frac{d y}{d x} = - \frac{F _{x}}{F _{y}}$