Tabla de derivadas y primitivas

Reglas de derivación

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 $u=C=\mathrm {cte}$

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

Regla de la cadena

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

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

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

Tabla de derivadas

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

Tabla de primitivas

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

Más primitivas en la Wikipedia

Series de Taylor

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

${\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\cdots$

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

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

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

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

$\mathrm {sen} (x)=x-{\frac {x^{3}}{6}}+{\frac {x^{5}}{5!}}+\cdots$

$\mathrm {arctg} (x)=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}+\cdots$