Na Galipedia, a Wikipedia en galego.
Na análise matemática , a regra do produto ou regra de Leibniz para a derivación dun produto , establece como derivar o produto de funcións derivables.
Pode declararse informalmente como: "A derivada dun produto de funcións é a derivada da primeira pola segunda sen derivar máis a derivada da segundo pola primeira sen derivar ". Matematicamente:
(
f
⋅
g
)
′
=
f
′
⋅
g
+
f
⋅
g
′
{\displaystyle (f\cdot g)'=f'\cdot g+f\cdot g'\,}
Ou usando a notación de Leibniz :
d
d
x
(
u
⋅
v
)
=
u
d
v
d
x
+
v
d
u
d
x
{\displaystyle {d \over dx}(u\cdot v)=u{dv \over dx}+v{du \over dx}}
Na resolución final de suma de produtos, a orde é indiferente, o importante é non confundir f(x), g(x), f'(x) e g'(x).
Pode chegarse á regra usando as características do límite e a definición da derivada como o límite do cociente da diferenza .
Entón, temos:
f
(
x
)
=
g
(
x
)
h
(
x
)
{\displaystyle f(x)=g(x)h(x)\,}
supoñendo que g e h son diferenciables na variable x . Logo
f
′
(
x
)
=
lim
Δ
x
→
0
f
(
x
+
Δ
x
)
−
f
(
x
)
Δ
x
=
lim
Δ
x
→
0
g
(
x
+
Δ
x
)
h
(
x
+
Δ
x
)
−
g
(
x
)
h
(
x
)
Δ
x
{\displaystyle f'(x)=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}=\lim _{\Delta x\to 0}{\frac {g(x+\Delta x)h(x+\Delta x)-g(x)h(x)}{\Delta x}}}
Como
g
(
x
+
Δ
x
)
h
(
x
+
Δ
x
)
−
g
(
x
)
h
(
x
)
=
g
(
x
)
(
h
(
x
+
Δ
x
)
−
h
(
x
)
)
+
h
(
x
+
Δ
x
)
(
g
(
x
+
Δ
x
)
−
g
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x
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)
,
{\displaystyle g\left(x+\Delta x\right)h(x+\Delta x)-g(x)h(x)=g(x)(h(x+\Delta x)-h(x))+h(x+\Delta x)(g(x+\Delta x)-g(x)),}
tense que
f
′
(
x
)
=
lim
Δ
x
→
0
g
(
x
+
Δ
x
)
h
(
x
+
Δ
x
)
−
g
(
x
)
h
(
x
)
+
g
(
x
)
h
(
x
+
Δ
x
)
−
g
(
x
)
h
(
x
+
Δ
x
)
Δ
x
{\displaystyle f'(x)=\lim _{\Delta x\to 0}{\frac {g(x+\Delta x)h(x+\Delta x)-g(x)h(x)+g(x)h(x+\Delta x)-g(x)h(x+\Delta x)}{\Delta x}}}
=
lim
Δ
x
→
0
g
(
x
)
(
h
(
x
+
Δ
x
)
−
h
(
x
)
)
+
h
(
x
+
Δ
x
)
(
g
(
x
+
Δ
x
)
−
g
(
x
)
)
Δ
x
{\displaystyle =\lim _{\Delta x\to 0}{\frac {g(x)(h(x+\Delta x)-h(x))+h(x+\Delta x)(g(x+\Delta x)-g(x))}{\Delta x}}}
=
lim
Δ
x
→
0
[
g
(
x
)
(
(
h
(
x
+
Δ
x
)
−
h
(
x
)
)
Δ
x
)
+
h
(
x
+
Δ
x
)
(
(
g
(
x
+
Δ
x
)
−
g
(
x
)
)
Δ
x
)
]
{\displaystyle =\lim _{\Delta x\to 0}\left[g(x)\left({\frac {(h(x+\Delta x)-h(x))}{\Delta x}}\right)+h(x+\Delta x)\left({\frac {(g(x+\Delta x)-g(x))}{\Delta x}}\right)\right]}
Como h é continua en x , tense que
lim
Δ
x
→
0
h
(
x
+
Δ
x
)
=
h
(
x
)
{\displaystyle \lim _{\Delta x\to 0}h(x+\Delta x)=h(x)}
e pola definición de derivada, e a diferenciabilidade de h e g en x , tense tamén que
[
lim
Δ
x
→
0
(
h
(
x
+
Δ
x
)
−
h
(
x
)
)
Δ
x
]
=
h
′
(
x
)
{\displaystyle \left[\lim _{\Delta x\to 0}{\frac {(h(x+\Delta x)-h(x))}{\Delta x}}\right]=h'(x)}
[
lim
Δ
x
→
0
(
g
(
x
+
Δ
x
)
−
g
(
x
)
)
Δ
x
]
=
g
′
(
x
)
{\displaystyle \left[\lim _{\Delta x\to 0}{\frac {(g(x+\Delta x)-g(x))}{\Delta x}}\right]=g'(x)}
Así, xustifícase a descomposición dos produtos dentro do límite, e reorganizando todo chégase á regra do produto.
f
′
(
x
)
=
lim
Δ
x
→
0
[
g
(
x
)
(
(
h
(
x
+
Δ
x
)
−
h
(
x
)
)
Δ
x
)
+
h
(
x
+
Δ
x
)
(
(
g
(
x
+
Δ
x
)
−
g
(
x
)
)
Δ
x
)
]
{\displaystyle f'(x)=\lim _{\Delta x\to 0}\left[g(x)\left({\frac {(h(x+\Delta x)-h(x))}{\Delta x}}\right)+h(x+\Delta x)\left({\frac {(g(x+\Delta x)-g(x))}{\Delta x}}\right)\right]}
=
[
lim
Δ
x
→
0
g
(
x
)
]
[
lim
Δ
x
→
0
(
h
(
x
+
Δ
x
)
−
h
(
x
)
)
Δ
x
]
+
[
lim
Δ
x
→
0
h
(
x
+
Δ
x
)
]
[
lim
Δ
x
→
0
(
g
(
x
+
Δ
x
)
−
g
(
x
)
)
Δ
x
]
{\displaystyle =\left[\lim _{\Delta x\to 0}g(x)\right]\left[\lim _{\Delta x\to 0}{\frac {(h(x+\Delta x)-h(x))}{\Delta x}}\right]+\left[\lim _{\Delta x\to 0}h(x+\Delta x)\right]\left[\lim _{\Delta x\to 0}{\frac {(g(x+\Delta x)-g(x))}{\Delta x}}\right]}
=
g
(
x
)
h
′
(
x
)
+
h
(
x
)
g
′
(
x
)
{\displaystyle =g(x)h'(x)+h(x)g'(x)\,}
Supoñendo que se quere derivar:
f
(
x
)
=
x
2
sin
(
x
)
{\displaystyle f(x)=x^{2}\,\sin(x)}
Usando a regra do produto, obtense a derivada:
f
′
(
x
)
=
2
x
sin
(
x
)
+
x
2
cos
(
x
)
{\displaystyle f^{\prime }(x)=2x\,\sin(x)+x^{2}\,\cos(x)}
xa que a derivada de
x
2
{\displaystyle x^{2}\,}
2
x
{\displaystyle 2x\,}
e a derivada de
sin
(
x
)
{\displaystyle \sin(x)\,}
cos
(
x
)
{\displaystyle \cos(x)\,}
Esta regra pode ser xeneralizada para a obtención do termo dunha derivación sucesiva de produto.
Sexan f e g funcións n-veces diferenciables. A derivada enésima do produto
f
⋅
g
{\displaystyle f\cdot g}
(
f
⋅
g
)
(
n
)
=
∑
k
=
0
n
(
n
k
)
f
(
k
)
g
(
n
−
k
)
{\displaystyle (f\cdot g)^{(n)}=\sum _{k=0}^{n}{n \choose k}f^{(k)}g^{(n-k)}}
onde
(
n
k
)
{\displaystyle {n \choose k}}
coeficiente binomial .
Isto próbase a través da regra do produto e a indución .
![{\displaystyle =\lim _{\Delta x\to 0}\left[g(x)\left({\frac {(h(x+\Delta x)-h(x))}{\Delta x}}\right)+h(x+\Delta x)\left({\frac {(g(x+\Delta x)-g(x))}{\Delta x}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a97b5870aa2eecae3a3644dcac9c4e407ef7188b)
![{\displaystyle \left[\lim _{\Delta x\to 0}{\frac {(h(x+\Delta x)-h(x))}{\Delta x}}\right]=h'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a33355ef9f825d1e20bae48f86f30c50944fc971)
![{\displaystyle \left[\lim _{\Delta x\to 0}{\frac {(g(x+\Delta x)-g(x))}{\Delta x}}\right]=g'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f84ff4c1c24330e72e8f22b452fa3df82d791d6)
![{\displaystyle f'(x)=\lim _{\Delta x\to 0}\left[g(x)\left({\frac {(h(x+\Delta x)-h(x))}{\Delta x}}\right)+h(x+\Delta x)\left({\frac {(g(x+\Delta x)-g(x))}{\Delta x}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03427fd70902afbe35aba6f8acdaee5d5c12c0f0)
![{\displaystyle =\left[\lim _{\Delta x\to 0}g(x)\right]\left[\lim _{\Delta x\to 0}{\frac {(h(x+\Delta x)-h(x))}{\Delta x}}\right]+\left[\lim _{\Delta x\to 0}h(x+\Delta x)\right]\left[\lim _{\Delta x\to 0}{\frac {(g(x+\Delta x)-g(x))}{\Delta x}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02cc708cdd89fdf2b56cf9f8af91fbede8dcb132)
