Common derivatives rules

Basic rules

$$ \frac{d}{dx}(- f(x)) = - \frac{d}{dx}(f(x)) \notag $$
$$ \frac{d}{dx}( f(x) + g(x) )= \frac{d}{dx} f(x) + \frac{d}{dx}g(x) \notag $$
$$ \frac{d}{dx}( f(x) - g(x) )= \frac{d}{dx} f(x) - \frac{d}{dx}g(x) \notag $$
$$ \frac{d}{dx}(c f(x)) = c \frac{d}{dx}(f(x)) \notag $$
$$ \frac{d}{dx}( x^n ) = nx^{n-1} \notag $$
$$ \frac{d}{dx}(c) = 0 \notag $$

Product rule

$$ (fg)' = f'g + fg' \notag $$

$$ \frac{d}{dx}(f(x) \times g(x)) = g(x) \times \frac{d}{dx}f(x) + f(x) \times \frac{d}{dx}g(x) \notag $$

Quotient rule

$$ \left( \frac {f}{g} \right) ' = \frac{f'g + fg'}{g^2} \notag $$

$$ \frac{d}{dx} \frac {f(x)}{g(x)} = \frac {g(x) \times \frac{d}{dx}f(x) + f(x) \times \frac{d}{dx}g(x)} {g^2(x)} \notag $$

Chain rule (composition of two functions)

$$ (f \circ g)'=(f' \circ g).g' \notag $$

$$ ( f(g(x)) )' = f'(g(x)).g'(x) \notag $$

$$ \frac{d}{dx} f(g(x)) = \frac{df}{dg}\frac{dg}{dx} \notag $$

Exponential

$$ \frac {d}{dx} \mathrm{e}^{ g(x) } = g'(x) \mathrm{e}^{g(x)} \notag $$

$$ \frac {d}{dx} \mathrm{e}^{ g(x) } = \frac{dg(x)}{dx} \times \mathrm{e}^{g(x)} \notag $$

Logarithm

$$ \frac {d}{dx} \ln (g(x)) = \frac {g'(x)}{g(x)} \notag $$

$$ \frac {d}{dx} \ln (g(x)) = \frac { \frac{d}{dx}g(x)}{g(x)} \notag $$

See also


Last update : 06/11/2018