Derivadas inmediatas
$f(x)=k \to f^{'}(x)=0$
$f(x)=x \to f^{'}(x)=1$
$f(x)=x^{n} \to f^{'}(x)=nx^{n-1}$
$f(x)=\sin x \to f^{'}(x)=\cos x$
$f(x)=\cos x \to f^{'}(x)=-\sin x$
$f(x)=\ln x \to f^{'}(x)=\dfrac{1}{x}$
$f(x)=\log_{a} x \to f^{'}(x)=\dfrac{1}{\ln a}·\dfrac{1}{x}$
$f(x)=e^{x} \to f^{'}(x)=e^{x}$
$f(x)=a^{x} \to f^{'}(x)=a^{x}\ln a$
$f(x)=\arcsin x \to f^{'}(x)=\dfrac{1}{\sqrt{1-x^{2}}}$
$f(x)=\arccos x \to f^{'}(x)=\dfrac{-1}{\sqrt{1-x^{2}}}$
$f(x)=\arctan x \to f^{'}(x)=\dfrac{1}{1+x^{2}}$
Derivadas de la composición de dos funciones
Utilizando la regla de la cadena:
$f(x)=g(u(x)) \to f^{'}(x)=g^{'}(u(x))·u^{'}(x)$
podemos calcular las derivadas de las funciones compuestas:
$f(x)=[u(x)]^{n} \to f^{'}(x)=n[u(x)]^{n-1}·u^{'}(x)$
$f(x)=\sin [u(x)] \to f^{'}(x)=\cos [u(x)]·u^{'}(x)$
$f(x)=\cos [u(x)] \to f^{'}(x)=-\sin [u(x)]·u^{'}(x)$
$f(x)=\ln [u(x)] \to f^{'}(x)=\dfrac{1}{u(x)}·u^{'}(x)$
$f(x)=\log_{a} [u(x)] \to f^{'}(x)=\dfrac{1}{\ln a}·\dfrac{1}{u(x)}·u^{'}(x)$
$f(x)=e^{u(x)} \to f^{'}(x)=e^{u(x)}·u^{'}(x)$
$f(x)=a^{u(x)} \to f^{'}(x)=a^{u(x)}·\ln a·u^{'}(x)$
$f(x)=\arcsin [u(x)] \to f^{'}(x)=\dfrac{1}{\sqrt{1-[u(x)]^{2}}}·u^{'}(x)$
$f(x)=\arccos [u(x)] \to f^{'}(x)=\dfrac{-1}{\sqrt{1-[u(x)]^{2}}}·u^{'}(x)$
$f(x)=\arctan [u(x)] \to f^{'}(x)=\dfrac{1}{1+[u(x)]^{2}}·u^{'}(x)$
Derivadas de sumas, productos, cocientes y potencias de dos funciones
$f(x)=g(x)+h(x) \to f^{'}(x)=g^{'}(x)+h^{'}(x)$
$f(x)=k·g(x) \to f^{'}(x)=k·g^{'}(x)$
$f(x)=g(x)·h(x) \to f^{'}(x)=g^{'}(x)·h(x)+g(x)·h^{'}(x)$
$f(x)=\dfrac{1}{g(x)} \to f^{'}(x)=\dfrac{-g^{'}(x)}{[g(x)]^{2}}$
$f(x)=\dfrac{g(x)}{h(x)} \to f^{'}(x)=\dfrac{g^{'}(x)·h(x)-g(x)·h^{'}(x)}{[h(x)]^{2}}$
$f(x)=g(x)^{h(x)} \to f^{'}(x)=g(x)^{h(x)}\bigg[h^{'}(x)·\ln g(x)+h(x)\dfrac{g^{'}(x)}{g(x)}\bigg]$
