Anexo:Derivadas

Linealidad

${\displaystyle \left({f+g}\right)'=f'+g'}$

${\displaystyle \left({f-g}\right)'=f'-g'}$

${\displaystyle \left({cf}\right)'=cf'}$

Regla del producto

${\displaystyle \left({fg}\right)'=f'.g+f.g'}$

Regla del cociente

${\displaystyle \left({f \over g}\right)'={f'.g-f.g' \over g^{2}},\qquad g\neq 0}$

Caso particular

${\displaystyle \left({\frac {1}{f}}\right)'={\frac {-f'}{f^{2}}},\qquad f\neq 0}$

Regla de la cadena

${\displaystyle (f\circ g)'=f'(g)g'}$

${\displaystyle {d \over dx}k=0}$

${\displaystyle {d \over dx}x=1}$

${\displaystyle {d \over dx}(cx)=c}$

${\displaystyle {d \over dx}x^{c}=cx^{c-1}\qquad {\mbox{donde }}x^{c}{\mbox{ y }}cx^{c-1}{\mbox{ se encuentran definidos}}}$

${\displaystyle {d \over dx}(cx^{n})=cnx^{n-1}}$

${\displaystyle {d \over dx}|x|={x \over |x|}=\operatorname {sgn} x,\qquad x\neq 0}$

${\displaystyle {d \over dx}\left({1 \over x}\right)={d \over dx}\left(x^{-1}\right)=-x^{-2}=-{1 \over x^{2}}}$

${\displaystyle {d \over dx}\left({1 \over x^{c}}\right)={d \over dx}\left(x^{-c}\right)=-cx^{-c-1}=-{c \over x^{c+1}}}$

${\displaystyle {d \over dx}({\sqrt[{n}]{x}})={1 \over n{\sqrt[{n}]{x^{n-1}}}}\,{\mbox{sea }}x>0}$

${\displaystyle {d \over dx}{\sqrt {x}}={d \over dx}x^{1 \over 2}={1 \over 2}x^{-{1 \over 2}}={1 \over 2{\sqrt {x}}},\qquad x>0}$

${\displaystyle {d \over dx}f(x)^{n}\ =nf(x)^{n-1}\cdot {d \over dx}f(x)}$

Derivada de la función inversa

${\displaystyle (f^{-1})'={\frac {1}{f'\circ f^{-1}}}}$,

para alguna función diferenciable f de un argumento real y con valores reales, cuando las composiciones indicadas e inversas existen.

${\displaystyle {d \over dx}c^{x}={c^{x}\ln c},\qquad c>0}$

${\displaystyle {d \over dx}e^{x}=e^{x}{d \over dx}(x)}$

${\displaystyle {d \over dx}\log _{c}x={1 \over x\ln c}\qquad ,c>0,c\neq 1}$

${\displaystyle {d \over dx}\ln x={1 \over x}\qquad ,x>0}$

${\displaystyle {d \over dx}\ln |x|={1 \over x}}$

${\displaystyle {d \over dx}x^{x}=x^{x}(1+\ln x)}$

${\displaystyle (f^{g})'=f^{g}\left(g'\ln f+{\frac {g}{f}}f'\right)}$

Derivada de la función potencial exponencial

${\displaystyle {d \over dx}f(x)^{g(x)}=f(x)^{g(x)}\left({d \over dx}f(x)\cdot {g(x) \over f(x)}+{d \over dx}g(x)\cdot \ln f(x)\right),\qquad f(x)>0}$

Derivadas trigonométricas cíclicas (Criterios de la primera, segunda y tercera derivadas)

La fórmula de Leibniz para diferenciación de integrales establece que:[3]

${\displaystyle {\frac {d}{dx}}\int _{g_{1}(x)}^{g_{2}(x)}f(x,s)\ ds=\int _{g_{1}(x)}^{g_{2}(x)}{\frac {\partial f(x,s)}{\partial x}}\ ds+f(x,g_{2}(x)){\frac {dg_{2}}{dx}}-f(x,g_{1}(x)){\frac {dg_{1}}{dx}}}$