Tabla de derivadas y primitivas

(Diferencias entre revisiones)

Revisión de 19:17 3 oct 2011

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

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

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

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

Más integrales en la Wikipedia

$(1+x)^n = 1 + n x + \frac{n(n-1)}{2}x^2+\frac{n(n-1)(n-2)}{3!}x^3+ \cdots$

$\frac{1}{1-x}=1+x+x^2+x^3+\cdots = \sum_{n=0}^\infty x^n$

$\frac{1}{(1-x)^2}=1+2x+3x^2+4x^3+\cdots = \sum_{n=0}^\infty (n+1)x^n$

$-\ln(1-x)=x+\frac{x^2}{2}+\frac{x^3}{3}+\cdots = \sum_{n=1}^\infty \frac{x^n}{n}$

$\mathrm{e}^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\cdots = \sum_{n=0}^\infty \frac{x^n}{n!}$

$cos(x)=1-\frac{x^2}{2}+\frac{x^4}{4!}+\cdots = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}$

$\mathrm{sen}(x)=x-\frac{x^3}{6}+\frac{x^5}{5!}+\cdots = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!}$

$\mathrm{arctg}(x)=x-\frac{x^3}{3}+\frac{x^5}{5}+\cdots = \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{2n+1}$

@@ Línea 85: / Línea 85: @@
 | <math>\ln|x|\,</math>
 | <math>\mathrm{tg}(x)\,</math>
-| <math>-\ln(\cos(x))\,</math>
+| <math>-\ln|\cos(x)|\,</math>
 |-
 | <math>x^n \quad (n\neq -1)</math>