Derivative rules cheat sheet

Differentiation is mechanical once you can name the structure: sum, product, quotient or composition. Identify the outermost structure first, then apply the matching rule.

Practise derivatives

Structural rules

$(c f)^{'} = c f^{'}$
$(f \pm g)^{'} = f^{'} \pm g^{'}$
$(f g)^{'} = f^{'} g + f g^{'}$ Product
$(\frac{f}{g})^{'} = \frac{f ^{'} g - f g ^{'}}{g ^{2}}$ Quotient
$\frac{d}{d x} f (g (x)) = f^{'} (g (x)) g^{'} (x)$ Chain

Core functions

$\frac{d}{d x} x^{n} = n x^{n - 1}$
$\frac{d}{d x} e^{x} = e^{x}$
$\frac{d}{d x} a^{x} = a^{x} ln a$
$\frac{d}{d x} ln x = \frac{1}{x}$
$\frac{d}{d x} lo g_{a} x = \frac{1}{x l n a}$

Trigonometric

$\frac{d}{d x} sin x = cos x$
$\frac{d}{d x} cos x = - sin x$
$\frac{d}{d x} tan x = sec^{2} x$
$\frac{d}{d x} sec x = sec x tan x$
$\frac{d}{d x} arcsin x = \frac{1}{1 - x ^{2}}$
$\frac{d}{d x} arctan x = \frac{1}{1 + x ^{2}}$

Applications

$y - f (a) = f^{'} (a) (x - a)$ Tangent line
$f^{'} (c) = 0 or undefined$ Critical points
$f^{''} > 0 \Rightarrow concave up$

Related calculators

Derivative Calculator