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Tabla de derivadas - Wikipedia, la enciclopedia libre

Tabla de derivadas

De Wikipedia, la enciclopedia libre

La operación fundamental en el cálculo diferencial es encontrar una derivada. Esta tabla enlista las derivadas de varias funciones. En lo sucesivo, f y g son funciones de x y c es una constante con respecto a x. Se presupone al conjunto de los números reales. Estas fórmulas son suficientes para diferenciar cualquier función elemental.

[editar] Reglas para la diferenciación de funciones generales

{d \over dx} c.v = c{d \over dx} v
{d \over dx} (f(x) + g(x)) = {d \over dx} f(x) + {d \over dx} g(x)
{d \over dx} (f(x) - g(x)) = {d \over dx} f(x) - {d \over dx} g(x)
{d \over dx} f(x)g(x) = {d \over dx}f(x) \cdot g(x) + f(x) \cdot {d \over dx}g(x)
{d \over dx} {f(x) \over g(x)} = {{d \over dx} f(x) \cdot g(x) - f(x) \cdot {d \over dx} g(x) \over (g(x))^2}
{d \over dx} f(x)^{g(x)} = f(x)^{g(x)}\left({d \over dx}f(x) \cdot {g(x) \over f(x)} + {d \over dx}g(x) \cdot \ln f(x)\right),\qquad f(x) > 0
{d \over dx} f(g(x)) = {d \over dg} f(g(x)) {d \over dx} g(x)

[editar] Derivadas de funciones simples

{d \over dx} c = 0
{d \over dx} x = 1
{d \over dx} |x| = {x \over |x|},\qquad x \ne 0
{d \over dx} x^c = cx^{c-1} demostración de x^n
{d \over dx} \sqrt{x} = {1 \over 2 \sqrt{x}}
{d \over dx} {1 \over x} = -{1 \over x^2}

Derivadas de funciones trigonométricas ===

{d \over dx} \rm{sen } x = \cos x
{d \over dx} \cos x = -\rm{sen } x
{d \over dx} \tan x = \sec^2 x
{d \over dx} \sec x = \tan x \sec x
{d \over dx} \cot x = -\csc^2 x
{d \over dx} \csc x = -\cot x \csc x
{d \over dx} \rm{arcsen } x = { 1 \over \sqrt{1 - x^2}}
{d \over dx} \arccos x = {-1 \over \sqrt{1 - x^2}}
{d \over dx} \arctan x = { 1 \over 1 + x^2}
{d \over dx} \arcsec x = { 1 \over |x|\sqrt{x^2 - 1}}
{d \over dx} \arccot x = {-1 \over 1 + x^2}
{d \over dx} \rm{arccosec } x = {-1 \over |x|\sqrt{x^2 - 1}}

[editar] Derivadas de funciones hiperbólicas

{d \over dx} \rm{senh } x = \cosh x
{d \over dx} \cosh x = \rm{senh } x
{d \over dx} \tanh x = \mbox{sech}^2\,x
{d \over dx} \,\mbox{sech}\,x = -\tanh x\,\mbox{sech}\,x
{d \over dx} \,\mbox{coth}\,x = -\,\mbox{csch}^2\,x
{d \over dx} \,\mbox{csch}\,x = -\,\mbox{coth}\,x\,\mbox{csch}\,x
{d \over dx} \rm{sinh }^{-1} x = { 1 \over \sqrt{x^2 + 1}}
{d \over dx} \cosh^{-1} x = {-1 \over \sqrt{x^2 - 1}}
{d \over dx} \tanh^{-1} x = { 1 \over 1 - x^2}
{d \over dx} \mbox{sech}^{-1}\,x = { 1 \over x\sqrt{1 - x^2}}
{d \over dx} \mbox{coth}^{-1}\,x = {-1 \over 1 - x^2}
{d \over dx} \mbox{csch}^{-1}\,x = {-1 \over |x|\sqrt{1 + x^2}}
Obtenido de "http://es.wikipedia.org../../../t/a/b/Tabla_de_derivadas.html"
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