Tabla de derivadas y primitivas

Contenido

Suma de funciones: $\frac{\mathrm{d}\ }{\mathrm{d}x}(u+v) = \frac{\mathrm{d}u}{\mathrm{d}x} + \frac{\mathrm{d}v}{\mathrm{d}x}$

Producto de funciones: $\frac{\mathrm{d}\ }{\mathrm{d}x}(uv) = \left(\frac{\mathrm{d}u}{\mathrm{d}x}\right)v + u\left(\frac{\mathrm{d}v}{\mathrm{d}x}\right)$

Caso particular $u = C = cte$

$\frac{\mathrm{d}\ }{\mathrm{d}x}(Cv) = C\left(\frac{\mathrm{d}v}{\mathrm{d}x}\right)$

$\frac{\mathrm{d}v}{\mathrm{d}x}=\frac{\mathrm{d}v}{\mathrm{d}u}\,\frac{\mathrm{d}u}{\mathrm{d}x}$

Caso particular de derivada logarítmica

$\frac{\mathrm{d}\ }{\mathrm{d}x}(\ln(u)) = \frac{1}{u}\,\frac{\mathrm{d}u}{\mathrm{d}x}$

Caso particular de exponencial de una función

$\frac{\mathrm{d}\ }{\mathrm{d}x}\left(\mathrm{e}^u\right) = \mathrm{e}^u\,\frac{\mathrm{d}u}{\mathrm{d}x}$

$f (x)$	$d f / d x$	$f (x)$	$d f / d x$
$C\,$	$0\,$	$\ln(x)\,$	$\frac{1}{x}$
$x\,$	$1\,$	$\mathrm{sen}(x)\,$	$\cos(x)\,$
$x^2\,$	$2x\,$	$\cos(x)\,$	$-\mathrm{sen}(x)\,$
$\frac{1}{x}$	$-\frac{1}{x^2}$	$\mathrm{tg}(x)\,$	$\frac{1}{\cos^2(x)}$
$x^n\,$	$nx^{n-1}\,$	$\mathrm{arcsen}(x)\,$	$\displaystyle\frac{1}{\sqrt{1-x^2}}$
$\mathrm{e}^x\,$	$\mathrm{e}^x\,$	$\mathrm{arccos}(x)\,$	$\displaystyle -\frac{1}{\sqrt{1-x^2}}$
$a^x\,$	$\ln(a)\,a^x$	$\mathrm{arctg}(x)\,$	$\displaystyle\frac{1}{1+x^2}$

$f (x)$	$\int f\,\mathrm{d}x$	$f (x)$	$\int f\,\mathrm{d}x$
$k\,$	$kx+C\,$	$\mathrm{sen}(x)\,$	$-\cos(x)+C\,$
$x\,$	$\frac{x^2}{2}+C\,$	$\cos(x)\,$	$\mathrm{sen}(x)+C\,$
$\frac{1}{x}\,$	$\ln\|x\|+C\,$	$\mathrm{tg}(x)\,$	$-\ln\|\cos(x)\|+C\,$
$x^n \quad (n\neq -1)$	$\frac{x^{n+1}}{n+1}+C$	$\frac{1}{\mathrm{sen}(x)}$	$\ln\left\|\mathrm{tg}\left(\frac{x}{2}\right)\right\|+C$
$\mathrm{e}^x\,$	$\mathrm{e}^x+C\,$	$\frac{1}{1+x^2}$	$\mathrm{arctg}(x)+C\,$
$\mathrm{e}^{\lambda x}\,$	$\frac{1}{\lambda}\mathrm{e}^{\lambda x}+C\,$	$\displaystyle\frac{1}{\sqrt{1-x^2}}$	$\mathrm{arcsen}(x)+C\,$