Tabla de derivadas

Referencia: Leon W Couch Apéndice p656

Tabla de Derivadas

Definición

$\frac{d}{dx}[ f(x) ]= \lim_{\Delta x \rightarrow\ 0} \frac{f \big( x+\frac{\Delta x}{2} \big)- f\big( x-\frac{\Delta x}{2}\big) }{\Delta x}$

Regla del producto

$\frac{d}{dx}[u(x) v(x)]= u(x)\frac{dv(x)}{dx} + v(x)\frac{du(x)}{dx}$

Regla del cociente

$\frac{d}{dx} \left[ \frac{u(x)}{v(x)}\right] = \frac{1}{v^2(x)} \left[ v(x)\frac{du(x)}{dx}- u(x)\frac{dv(x)}{dx}\right]$

Regla de la cadena

$\frac{d}{dx}u[v(x)]= \frac{du}{dv}\frac{dv}{dx}$

Potenciación

$\frac{d}{dx}[ x^n ]= nx^{n-1}$

Exponenciales

$\frac{d}{dx} [ e ^{ax} ] = a e^{ax}$ $\frac{d}{dx} [ a ^{x} ]= a^x ln(a)$

Logaritmicas

$\frac{d}{dx} [ ln(x) ] = \frac{1}{x}$ $\frac{d}{dx}[ log_a (x) ] = \frac{1}{x} log_a e$

Trigonométricas

$\frac{d}{dx}[sen(ax)]= a\text{ } cos(ax)$ $\frac{d}{dx}[cos(ax)]= -a\text{ }sen(ax)$ $\frac{d}{dx}[tan(ax)]= \frac{a} {cos^2(ax)}$ $\frac{d}{dx}[sen^{-1}(ax)]= \frac{a} {\sqrt{1-(ax)^2}}$ $\frac{d}{dx}[cos^{-1}(ax)]= \frac{-a} {\sqrt{1-(ax)^2}}$ $\frac{d}{dx}[tan^{-1}(ax)]= \frac{a} {{1+(ax)^2}}$

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Regla de Leibniz

$\frac{d}{dx}\left[ \int_{a(x)}^{b(x)} f(\lambda,x) d\lambda \right] =$
$= f(b(x),x) \frac{d}{dx}[b(x)] - f(a(x),x) \frac{d}{dx}[a(x)] +$
$+ \int_{a(x)}^{b(x)} \frac{d}{dx}[f(\lambda , x) d\lambda$

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