$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$$

Función	Derivada
$f(x)+g(x)$	$f^{\prime }(x)+g^{\prime }(x)$
$f(x)-g(x)$	$f^{\prime }(x)-g^{\prime }(x)$
$f(x)\cdot g(x)$	$f^{\prime }(x)\cdot g(x)+f(x)\cdot g^{\prime }(x)$
$\frac{f(x)}{g(x)}$	$\frac{f^{\prime }(x)\cdot g(x)-f(x)\cdot g^{\prime }(x)}{g(x{)}^2}$
$f(g(x))$	$f^{\prime }(g(x))\cdot g^{\prime }(x)$
$k:k\in \mathbb{R}$	$0$
$k\cdot f(x):k\in \mathbb{R}$	$k\cdot f^{\prime }(x)$
$x^r:r\in \mathbb{Q}$	$n{x}^{n-1}$
$f^{-1}(b)$	$\frac{1}{f^{\prime }(a)}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\frac{1}{{\cos }^2(x)}$
$\sec x$	$\frac{\sin x}{{\cos }^{2}x}$
$\mathrm{cosec}x$	$\frac{-\cos x}{{\sin }^{2}x}$
$\mathrm{cotg}x$	$\frac{-1}{{\sin }^{2}x}$
$\arcsin x$	$\frac{1}{\sqrt{1-x^2}}$
$\arccos x$	$\frac{-1}{\sqrt{1-x^2}}$
$\arctan x$	$\frac{1}{1+x^2}$
$a^x:a\in (0,1)\cup (1,+\infty )$	$a^x\cdot \ln a$
$\ln x$	$\frac{1}{x}$
$\cosh x$	$\sinh x$
$\sinh x$	$\cosh x$