Este es un resumen de reglas de diferenciación , esto es, reglas para calcular la derivado de una función en cálculo .
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A menos que se diga lo contrario, todas las funciones son funciones de números reales (
R
{\displaystyle \mathbb {R} }
f
:
R
→
R
{\displaystyle f:\mathbb {R} \to \mathbb {R} }
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Para cualesquiera funciones
f
{\displaystyle f}
g
{\displaystyle g}
a
{\displaystyle a}
b
{\displaystyle b}
h
(
x
)
=
a
f
(
x
)
+
b
g
(
x
)
{\displaystyle h(x)=af(x)+bg(x)}
x
{\displaystyle x}
h
′
(
x
)
=
a
f
′
(
x
)
+
b
g
′
(
x
)
.
{\displaystyle h'(x)=af'(x)+bg'(x).}
en la notación de Leibniz esto se escribe como:
d
(
a
f
+
b
g
)
d
x
=
a
d
f
d
x
+
b
d
g
d
x
.
{\displaystyle {\frac {d(af+bg)}{dx}}=a{\frac {df}{dx}}+b{\frac {dg}{dx}}.}
Casos especiales incluyen:
La regla del producto por una constante
(
a
f
)
′
=
a
f
′
{\displaystyle (af)'=af'}
(
f
+
g
)
′
=
f
′
+
g
′
{\displaystyle (f+g)'=f'+g'}
(
f
−
g
)
′
=
f
′
−
g
′
.
{\displaystyle (f-g)'=f'-g'.}
Para las funciones
f
{\displaystyle f}
g
{\displaystyle g}
h
(
x
)
=
f
(
x
)
g
(
x
)
{\displaystyle h(x)=f(x)g(x)}
x
{\displaystyle x}
h
′
(
x
)
=
(
f
g
)
′
(
x
)
=
f
′
(
x
)
g
(
x
)
+
f
(
x
)
g
′
(
x
)
.
{\displaystyle h'(x)=(fg)'(x)=f'(x)g(x)+f(x)g'(x).}
En la notación de Leibniz esto se escribe como
d
(
f
g
)
d
x
=
d
f
d
x
g
+
f
d
g
d
x
.
{\displaystyle {\frac {d(fg)}{dx}}={\frac {df}{dx}}g+f{\frac {dg}{dx}}.}
La derivada de la función
h
(
x
)
=
f
(
g
(
x
)
)
{\displaystyle h(x)=f(g(x))}
h
′
(
x
)
=
f
′
(
g
(
x
)
)
⋅
g
′
(
x
)
.
{\displaystyle h'(x)=f'(g(x))\cdot g'(x).}
En la notación de Leibniz esto se escribe como:
d
d
x
h
(
x
)
=
d
d
z
f
(
z
)
|
z
=
g
(
x
)
⋅
d
d
x
g
(
x
)
,
{\displaystyle {\frac {d}{dx}}h(x)={\frac {d}{dz}}f(z)|_{z=g(x)}\cdot {\frac {d}{dx}}g(x),}
a menudo abreviado a
d
h
(
x
)
d
x
=
d
f
(
g
(
x
)
)
d
g
(
x
)
⋅
d
g
(
x
)
d
x
.
{\displaystyle {\frac {dh(x)}{dx}}={\frac {df(g(x))}{dg(x)}}\cdot {\frac {dg(x)}{dx}}.}
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Si la función
f
{\displaystyle f}
función inversa
g
{\displaystyle g}
g
(
f
(
x
)
)
=
x
{\displaystyle g(f(x))=x}
f
(
g
(
y
)
)
=
y
{\displaystyle f(g(y))=y}
g
′
=
1
f
′
∘
g
.
{\displaystyle g'={\frac {1}{f'\circ g}}.}
En Leibniz notación esto se escribe como
d
x
d
y
=
1
d
y
d
x
.
{\displaystyle {\frac {dx}{dy}}={\frac {1}{\frac {dy}{dx}}}.}
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editar
d
d
x
(
c
a
x
)
=
a
c
a
x
ln
c
,
c
>
0
{\displaystyle {\frac {d}{dx}}\left(c^{ax}\right)={ac^{ax}\ln c},\qquad c>0}
la ecuación de arriba es válida para todo
c
{\displaystyle c}
c
<
0
{\textstyle c<0}
número complejo .
d
d
x
(
e
a
x
)
=
a
e
a
x
{\displaystyle {\frac {d}{dx}}\left(e^{ax}\right)=ae^{ax}}
d
d
x
(
log
c
x
)
=
1
x
ln
c
,
c
>
0
,
c
≠
1
{\displaystyle {\frac {d}{dx}}\left(\log _{c}x\right)={1 \over x\ln c},\qquad c>0,c\neq 1}
la ecuación de arriba también es válida para todo
c
{\displaystyle c}
c
<
0
{\textstyle c<0\!}
d
d
x
(
ln
x
)
=
1
x
,
x
>
0.
{\displaystyle {\frac {d}{dx}}\left(\ln x\right)={1 \over x},\qquad x>0.}
d
d
x
(
ln
|
x
|
)
=
1
x
,
x
≠
0.
{\displaystyle {\frac {d}{dx}}\left(\ln |x|\right)={1 \over x},\qquad x\neq 0.}
d
d
x
(
x
x
)
=
x
x
(
1
+
ln
x
)
.
{\displaystyle {\frac {d}{dx}}\left(x^{x}\right)=x^{x}(1+\ln x).}
d
d
x
(
f
(
x
)
g
(
x
)
)
=
g
(
x
)
f
(
x
)
g
(
x
)
−
1
d
f
d
x
+
f
(
x
)
g
(
x
)
ln
(
f
(
x
)
)
d
g
d
x
,
if
f
(
x
)
>
0
,
y si
d
f
d
x
y
d
g
d
x
existen.
{\displaystyle {\frac {d}{dx}}\left(f(x)^{g(x)}\right)=g(x)f(x)^{g(x)-1}{\frac {df}{dx}}+f(x)^{g(x)}\ln {(f(x))}{\frac {dg}{dx}},\qquad {\text{if }}f(x)>0,{\text{ y si }}{\frac {df}{dx}}{\text{ y }}{\frac {dg}{dx}}{\text{ existen.}}}
d
d
x
(
f
1
(
x
)
f
2
(
x
)
(
.
.
.
)
f
n
(
x
)
)
=
[
∑
k
=
1
n
∂
∂
x
k
(
f
1
(
x
1
)
f
2
(
x
2
)
(
.
.
.
)
f
n
(
x
n
)
)
]
|
x
1
=
x
2
=
.
.
.
=
x
n
=
x
,
si
f
i
<
n
(
x
)
>
0
y
{\displaystyle {\frac {d}{dx}}\left(f_{1}(x)^{f_{2}(x)^{\left(...\right)^{f_{n}(x)}}}\right)=\left[\sum \limits _{k=1}^{n}{\frac {\partial }{\partial x_{k}}}\left(f_{1}(x_{1})^{f_{2}(x_{2})^{\left(...\right)^{f_{n}(x_{n})}}}\right)\right]{\biggr \vert }_{x_{1}=x_{2}=...=x_{n}=x},{\text{ si }}f_{i<n}(x)>0{\text{ y }}}
d
f
i
d
x
existe.
{\displaystyle {\frac {df_{i}}{dx}}{\text{ existe. }}}
La derivada logarítmica es otra manera de enunciar la regla para derivar el logaritmo de una función (utilizando la regla de cadena):
(
ln
f
)
′
=
f
′
f
{\displaystyle (\ln f)'={\frac {f'}{f}}\quad }
cuando
f
{\displaystyle f}
La diferenciación logarítmica es una técnica que utiliza logaritmos y sus reglas de diferenciación para simplificar ciertas expresiones antes de aplicar la derivada. Los logaritmos pueden ser utilizados para remover exponentes, convertir productos en sumas y convertir una división a una resta.
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(
sin
x
)
′
=
cos
x
{\displaystyle (\sin x)'=\cos x}
(
arcsin
x
)
′
=
1
1
−
x
2
{\displaystyle (\arcsin x)'={1 \over {\sqrt {1-x^{2}}}}}
(
cos
x
)
′
=
−
sin
x
{\displaystyle (\cos x)'=-\sin x}
(
arccos
x
)
′
=
−
1
1
−
x
2
{\displaystyle (\arccos x)'=-{1 \over {\sqrt {1-x^{2}}}}}
(
tan
x
)
′
=
sec
2
x
=
1
cos
2
x
=
1
+
tan
2
x
{\displaystyle (\tan x)'=\sec ^{2}x={1 \over \cos ^{2}x}=1+\tan ^{2}x}
(
arctan
x
)
′
=
1
1
+
x
2
{\displaystyle (\arctan x)'={1 \over 1+x^{2}}}
(
cot
x
)
′
=
−
csc
2
x
=
−
1
sin
2
x
=
−
(
1
+
cot
2
x
)
{\displaystyle (\cot x)'=-\csc ^{2}x=-{1 \over \sin ^{2}x}=-(1+\cot ^{2}x)}
(
arccot
x
)
′
=
−
1
1
+
x
2
{\displaystyle (\operatorname {arccot} x)'=-{1 \over 1+x^{2}}}
(
sec
x
)
′
=
tan
x
sec
x
{\displaystyle (\sec x)'=\tan x\sec x}
(
arcsec
x
)
′
=
1
|
x
|
x
2
−
1
{\displaystyle (\operatorname {arcsec} x)'={1 \over |x|{\sqrt {x^{2}-1}}}}
(
csc
x
)
′
=
−
cot
x
csc
x
{\displaystyle (\csc x)'=-\cot x\csc x}
(
arccsc
x
)
′
=
−
1
|
x
|
x
2
−
1
{\displaystyle (\operatorname {arccsc} x)'=-{1 \over |x|{\sqrt {x^{2}-1}}}}
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(
sinh
x
)
′
=
cosh
x
=
e
x
+
e
−
x
2
{\displaystyle (\sinh x)'=\cosh x={\frac {e^{x}+e^{-x}}{2}}}
(
arsinh
x
)
′
=
1
x
2
+
1
{\displaystyle (\operatorname {arsinh} \,x)'={1 \over {\sqrt {x^{2}+1}}}}
(
cosh
x
)
′
=
sinh
x
=
e
x
−
e
−
x
2
{\displaystyle (\cosh x)'=\sinh x={\frac {e^{x}-e^{-x}}{2}}}
(
arcosh
x
)
′
=
1
x
2
−
1
{\displaystyle (\operatorname {arcosh} \,x)'={\frac {1}{\sqrt {x^{2}-1}}}}
(
tanh
x
)
′
=
sech
2
x
{\displaystyle (\tanh x)'={\operatorname {sech} ^{2}\,x}}
(
artanh
x
)
′
=
1
1
−
x
2
{\displaystyle (\operatorname {artanh} \,x)'={1 \over 1-x^{2}}}
(
coth
x
)
′
=
−
csch
2
x
{\displaystyle (\operatorname {coth} \,x)'=-\,\operatorname {csch} ^{2}\,x}
(
arcoth
x
)
′
=
1
1
−
x
2
{\displaystyle (\operatorname {arcoth} \,x)'={1 \over 1-x^{2}}}
(
sech
x
)
′
=
−
tanh
x
sech
x
{\displaystyle (\operatorname {sech} \,x)'=-\tanh x\,\operatorname {sech} \,x}
(
arsech
x
)
′
=
−
1
x
1
−
x
2
{\displaystyle (\operatorname {arsech} \,x)'=-{1 \over x{\sqrt {1-x^{2}}}}}
(
csch
x
)
′
=
−
coth
x
csch
x
{\displaystyle (\operatorname {csch} \,x)'=-\,\operatorname {coth} \,x\,\operatorname {csch} \,x}
(
arcsch
x
)
′
=
−
1
|
x
|
1
+
x
2
{\displaystyle (\operatorname {arcsch} \,x)'=-{1 \over |x|{\sqrt {1+x^{2}}}}}
Derivadas de funciones especiales
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Función de Zeta del Riemann
ζ
(
x
)
=
∑
n
=
1
∞
1
n
x
{\displaystyle \quad \zeta (x)=\sum _{n=1}^{\infty }{\frac {1}{n^{x}}}}
ζ
′
(
x
)
=
−
∑
n
=
1
∞
ln
n
n
x
=
−
ln
2
2
x
−
ln
3
3
x
−
ln
4
4
x
−
⋯
{\displaystyle \zeta '(x)=-\sum _{n=1}^{\infty }{\frac {\ln n}{n^{x}}}=-{\frac {\ln 2}{2^{x}}}-{\frac {\ln 3}{3^{x}}}-{\frac {\ln 4}{4^{x}}}-\cdots }
=
−
∑
p
prime
p
−
x
ln
p
(
1
−
p
−
x
)
2
∏
q
prime
,
q
≠
p
1
1
−
q
−
x
{\displaystyle \,=-\sum _{p{\text{ prime}}}{\frac {p^{-x}\ln p}{(1-p^{-x})^{2}}}\prod _{q{\text{ prime}},q\neq p}{\frac {1}{1-q^{-x}}}}
Derivadas de integrales
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Supone que se requiere derivar con respetar a
x
{\displaystyle x}
F
(
x
)
=
∫
a
(
x
)
b
(
x
)
f
(
x
,
t
)
d
t
,
{\displaystyle F(x)=\int _{a(x)}^{b(x)}f(x,t)\,dt,}
donde las funciones
f
(
x
,
t
)
{\displaystyle f(x,t)}
∂
∂
x
f
(
x
,
t
)
{\displaystyle {\frac {\partial }{\partial x}}\,f(x,t)}
t
{\displaystyle t}
x
{\displaystyle x}
(
t
,
x
)
{\displaystyle (t,x)}
a
(
x
)
≤
t
≤
b
(
x
)
,
{\displaystyle a(x)\leq t\leq b(x),}
x
0
≤
x
≤
x
1
{\displaystyle x_{0}\leq x\leq x_{1}}
a
(
x
)
{\displaystyle a(x)}
b
(
x
)
{\displaystyle b(x)}
x
0
≤
x
≤
x
1
{\displaystyle x_{0}\leq x\leq x_{1}}
x
0
≤
x
≤
x
1
{\displaystyle \,x_{0}\leq x\leq x_{1}}
F
′
(
x
)
=
f
(
x
,
b
(
x
)
)
b
′
(
x
)
−
f
(
x
,
a
(
x
)
)
a
′
(
x
)
+
∫
a
(
x
)
b
(
x
)
∂
∂
x
f
(
x
,
t
)
d
t
.
{\displaystyle F'(x)=f(x,b(x))\,b'(x)-f(x,a(x))\,a'(x)+\int _{a(x)}^{b(x)}{\frac {\partial }{\partial x}}\,f(x,t)\;dt\,.}
esta fórmula es la forma general de la regla de diferenciación de Leibniz y puede ser obtenida utilizando el teorema fundamental de cálculo .
n
{\displaystyle n}
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![{\displaystyle {\frac {d}{dx}}\left(f_{1}(x)^{f_{2}(x)^{\left(...\right)^{f_{n}(x)}}}\right)=\left[\sum \limits _{k=1}^{n}{\frac {\partial }{\partial x_{k}}}\left(f_{1}(x_{1})^{f_{2}(x_{2})^{\left(...\right)^{f_{n}(x_{n})}}}\right)\right]{\biggr \vert }_{x_{1}=x_{2}=...=x_{n}=x},{\text{ si }}f_{i<n}(x)>0{\text{ y }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08db7359fb85a06886f634b9ac6a2967faee8c88)
![{\displaystyle {\frac {d^{n}}{dx^{n}}}[f(g(x))]=n!\sum _{\{k_{m}\}}^{}f^{(r)}(g(x))\prod _{m=1}^{n}{\frac {1}{k_{m}!}}\left(g^{(m)}(x)\right)^{k_{m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6db9fc6a48b3658dc330625f3c027300228b82b0)
![{\displaystyle {\frac {d^{n}}{dx^{n}}}[f(x)g(x)]=\sum _{k=0}^{n}{\binom {n}{k}}{\frac {d^{n-k}}{dx^{n-k}}}f(x){\frac {d^{k}}{dx^{k}}}g(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63405be1615e41e5620d8b0c45f48b7dd7e6f2ef)
Datos: Q1079958