Regla de la Cadena de Funciones de Varias Variables Angel Carrillo Hoyo, Elena de Oteyza de Oteyza\(^2\), Carlos Hernández Garciadiego\(^1\), Emma Lam Osnaya\(^2\) \(^1\) Instituto de Matemáticas, UNAM; \(^2\) Facultad de Ciencias, UNAM

Si \(z=f\left( x,y\right) \) es una función real de clase \(C^{2}\) en \(\mathbb{R}^{2}.\) Calcula \(\dfrac{\partial ^{2}z}{\partial r^{2}},\) \( \dfrac{\partial ^{2}z}{\partial s\partial r},\) \(\dfrac{\partial ^{2}z}{ \partial s^{2}} \) si \(x=\ln \left( r^{2}+rs\right) ,\) \(y=\ln \left( s^{2}+rs\right) .\)

Solución:

Observamos el siguiente diagrama

La función \(f\) depende de \(x\) y \(y;\) tanto \(x\) como \(y\) dependen de \(r\) y \(s.\)

Calculamos

\begin{eqnarray*} \dfrac{\partial z}{\partial r} & = &\dfrac{\partial f}{\partial x}\dfrac{ \partial x}{\partial r}+\dfrac{\partial f}{\partial y}\dfrac{\partial y}{ \partial r} \\ & = &\dfrac{\partial f}{\partial x}\left( \dfrac{2r+s}{r^{2}+rs}\right) + \dfrac{\partial f}{\partial y}\left( \dfrac{s}{s^{2}+rs}\right) \\ & = &\dfrac{2r+s}{r^{2}+rs}\dfrac{\partial f}{\partial x}+\dfrac{1}{s+r}\dfrac{ \partial f}{\partial y} \end{eqnarray*}

y

\begin{eqnarray*} \dfrac{\partial z}{\partial s} & = &\dfrac{\partial f}{\partial x}\dfrac{ \partial x}{\partial s}+\dfrac{\partial f}{\partial y}\dfrac{\partial y}{ \partial s} \\ & = &\dfrac{\partial f}{\partial x}\left( \dfrac{r}{r^{2}+rs}\right) +\dfrac{ \partial f}{\partial y}\left( \dfrac{2s+r}{s^{2}+rs}\right) \\ & = &\dfrac{1}{r+s}\dfrac{\partial f}{\partial x}+\dfrac{2s+r}{s^{2}+rs}\dfrac{ \partial f}{\partial y} \end{eqnarray*}

Ahora consideramos el siguiente diagrama

donde observamos que tanto \(\dfrac{ \partial f}{\partial x}\) como \(\dfrac{\partial f}{\partial y}\) dependen de \( x \) y \(y\) que a su vez dependen de \(r\) y \(s.\)

Calculamos

\begin{eqnarray*} \dfrac{\partial ^{2}z}{\partial r^{2}} & = &\dfrac{\partial }{\partial r} \left( \dfrac{\partial z}{\partial r}\right) \\ & = &\dfrac{\partial }{\partial r}\left( \dfrac{2r+s}{r^{2}+rs}\dfrac{\partial f}{\partial x}+\dfrac{1}{s+r}\dfrac{\partial f}{\partial y}\right) \\ & = &\dfrac{\partial }{\partial r}\left( \dfrac{2r+s}{r^{2}+rs}\right) \dfrac{ \partial f}{\partial x}+\dfrac{2r+s}{r^{2}+rs}\dfrac{\partial }{\partial r} \left( \dfrac{\partial f}{\partial x}\right) +\dfrac{\partial }{\partial r} \left( \dfrac{1}{s+r}\right) \dfrac{\partial f}{\partial y}+\dfrac{1}{s+r} \dfrac{\partial }{\partial r}\left( \dfrac{\partial f}{\partial y}\right) \\ & = &\dfrac{2\left( r^{2}+rs\right) -\left( 2r+s\right) \left( 2r+s\right) }{ \left( r^{2}+rs\right) ^{2}}\dfrac{\partial f}{\partial x}+\dfrac{2r+s}{ r^{2}+rs}\left( \dfrac{\partial }{\partial x}\left( \dfrac{\partial f}{ \partial x}\right) \dfrac{\partial x}{\partial r}+\dfrac{\partial }{\partial y}\left( \dfrac{\partial f}{\partial x}\right) \dfrac{\partial y}{\partial r} \right) -\dfrac{1}{\left( s+r\right) ^{2}}\dfrac{\partial f}{\partial y}+ \\ & &+\dfrac{1}{s+r}\left( \dfrac{\partial }{\partial x}\left( \dfrac{\partial f}{ \partial y}\right) \dfrac{\partial x}{\partial r}+\dfrac{\partial }{\partial y}\left( \dfrac{\partial f}{\partial y}\right) \dfrac{\partial y}{\partial r} \right) \\ & = &\dfrac{2\left( r^{2}+rs\right) -\left( 2r+s\right) \left( 2r+s\right) }{ \left( r^{2}+rs\right) ^{2}}\dfrac{\partial f}{\partial x}+\dfrac{2r+s}{ r^{2}+rs}\left( \dfrac{\partial ^{2}f}{\partial x^{2}}\dfrac{\partial x}{ \partial r}+\dfrac{\partial ^{2}f}{\partial y\partial x}\dfrac{\partial y}{ \partial r}\right) -\dfrac{1}{\left( s+r\right) ^{2}}\dfrac{\partial f}{ \partial y}+\dfrac{1}{s+r}\left( \dfrac{\partial ^{2}f}{\partial x\partial y} \dfrac{\partial x}{\partial r}+\dfrac{\partial ^{2}f}{\partial y^{2}}\dfrac{ \partial y}{\partial r}\right) \\ & = &\dfrac{-2r^{2}-2rs-s^{2}}{\left( r^{2}+rs\right) ^{2}}\dfrac{\partial f}{ \partial x}+\dfrac{2r+s}{r^{2}+rs}\left( \dfrac{\partial ^{2}f}{\partial x^{2}}\dfrac{2r+s}{r^{2}+rs}+\dfrac{\partial ^{2}f}{\partial y\partial x} \dfrac{1}{s+r}\right) -\dfrac{1}{\left( s+r\right) ^{2}}\dfrac{\partial f}{ \partial y}+\dfrac{1}{s+r}\left( \dfrac{\partial ^{2}f}{\partial x\partial y} \dfrac{2r+s}{r^{2}+rs}+\dfrac{\partial ^{2}f}{\partial y^{2}}\dfrac{1}{s+r} \right) \\ & = &\dfrac{-2r^{2}-2rs-s^{2}}{\left( r^{2}+rs\right) ^{2}}\dfrac{\partial f}{ \partial x}+\dfrac{\left( 2r+s\right) ^{2}}{\left( r^{2}+rs\right) ^{2}} \dfrac{\partial ^{2}f}{\partial x^{2}}+\dfrac{2r+s}{\left( r^{2}+rs\right) \left( s+r\right) }\dfrac{\partial ^{2}f}{\partial y\partial x}-\dfrac{1}{ \left( s+r\right) ^{2}}\dfrac{\partial f}{\partial y}+\dfrac{2r+s}{\left( s+r\right) \left( r^{2}+rs\right) }\dfrac{\partial ^{2}f}{\partial x\partial y}+\dfrac{1}{\left( s+r\right) ^{2}}\dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &\dfrac{-2r^{2}-2rs-s^{2}}{\left( r^{2}+rs\right) ^{2}}\dfrac{\partial f}{ \partial x}+\dfrac{\left( 2r+s\right) ^{2}}{\left( r^{2}+rs\right) ^{2}} \dfrac{\partial ^{2}f}{\partial x^{2}}+\dfrac{2\left( 2r+s\right) }{\left( r^{2}+rs\right) \left( s+r\right) }\dfrac{\partial ^{2}f}{\partial y\partial x}-\dfrac{1}{\left( s+r\right) ^{2}}\dfrac{\partial f}{\partial y}+\dfrac{1}{ \left( s+r\right) ^{2}}\dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}

La derivada de segundo orden mixta es

\begin{eqnarray*} \dfrac{\partial ^{2}z}{\partial s\partial r} & = & \dfrac{\partial }{\partial s} \left( \dfrac{\partial z}{\partial r}\right) \\ & = & \dfrac{\partial }{\partial s}\left( \dfrac{2r+s}{r^{2}+rs}\dfrac{\partial f}{\partial x}+\dfrac{1}{s+r}\dfrac{\partial f}{\partial y}\right) \\ &=&\dfrac{\partial }{\partial s}\left( \dfrac{2r+s}{r^{2}+rs}\right) \dfrac{ \partial f}{\partial x}+\dfrac{2r+s}{r^{2}+rs}\dfrac{\partial }{\partial s} \left( \dfrac{\partial f}{\partial x}\right) +\dfrac{\partial }{\partial s} \left( \dfrac{1}{s+r}\right) \dfrac{\partial f}{\partial y}+\dfrac{1}{s+r} \dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{\partial y}\right) \\ & = & \dfrac{r^{2}+rs-r\left( 2r+s\right) }{\left( r^{2}+rs\right) ^{2}}\dfrac{ \partial f}{\partial x}+\dfrac{2r+s}{r^{2}+rs}\left( \dfrac{\partial }{ \partial x}\left( \dfrac{\partial f}{\partial x}\right) \dfrac{\partial x}{ \partial s}+\dfrac{\partial }{\partial y}\left( \dfrac{\partial f}{\partial x }\right) \dfrac{\partial y}{\partial s}\right) -\dfrac{1}{\left( s+r\right) ^{2}}\dfrac{\partial f}{\partial y}+\dfrac{1}{s+r}\left( \dfrac{\partial }{ \partial x}\left( \dfrac{\partial f}{\partial y}\right) \dfrac{\partial x}{ \partial s}+\dfrac{\partial }{\partial y}\left( \dfrac{\partial f}{\partial y }\right) \dfrac{\partial y}{\partial s}\right) \\ & = & \dfrac{r^{2}+rs-r\left( 2r+s\right) }{\left( r^{2}+rs\right) ^{2}}\dfrac{ \partial f}{\partial x}+\dfrac{2r+s}{r^{2}+rs}\left( \dfrac{\partial ^{2}f}{ \partial x^{2}}\dfrac{\partial x}{\partial s}+\dfrac{\partial ^{2}f}{ \partial y\partial x}\dfrac{\partial y}{\partial s}\right) -\dfrac{1}{\left( s+r\right) ^{2}}\dfrac{\partial f}{\partial y}+\dfrac{1}{s+r}\left( \dfrac{ \partial ^{2}f}{\partial x\partial y}\dfrac{\partial x}{\partial s}+\dfrac{ \partial ^{2}f}{\partial y^{2}}\dfrac{\partial y}{\partial s}\right) \\ & = & -\dfrac{r^{2}}{\left( r^{2}+rs\right) ^{2}}\dfrac{\partial f}{\partial x}+ \dfrac{2r+s}{r^{2}+rs}\left( \dfrac{\partial ^{2}f}{\partial x^{2}}\dfrac{1}{ r+s}+\dfrac{\partial ^{2}f}{\partial y\partial x}\dfrac{2s+r}{s^{2}+rs} \right) -\dfrac{1}{\left( s+r\right) ^{2}}\dfrac{\partial f}{\partial y}+ \dfrac{1}{s+r}\left( \dfrac{\partial ^{2}f}{\partial x\partial y}\dfrac{1}{ r+s}+\dfrac{\partial ^{2}f}{\partial y^{2}}\dfrac{2s+r}{s^{2}+rs}\right) \\ & = & -\dfrac{r^{2}}{\left( r^{2}+rs\right) ^{2}}\dfrac{\partial f}{\partial x}+ \dfrac{2r+s}{\left( r^{2}+rs\right) \left( r+s\right) }\dfrac{\partial ^{2}f }{\partial x^{2}}+\dfrac{\left( 2r+s\right) \left( 2s+r\right) }{\left( r^{2}+rs\right) \left( s^{2}+rs\right) }\dfrac{\partial ^{2}f}{\partial y\partial x}-\dfrac{1}{\left( s+r\right) ^{2}}\dfrac{\partial f}{\partial y}+ \dfrac{1}{\left( s+r\right) ^{2}}\dfrac{\partial ^{2}f}{\partial x\partial y} +\dfrac{2s+r}{\left( s^{2}+rs\right) \left( s+r\right) }\dfrac{\partial ^{2}f }{\partial y^{2}} \\ & = & -\dfrac{1}{\left( r+s\right) ^{2}}\dfrac{\partial f}{\partial x}+\dfrac{ 2r+s}{r\left( r+s\right) ^{2}}\dfrac{\partial ^{2}f}{\partial x^{2}}+\dfrac{ \left( 2r+s\right) \left( 2s+r\right) }{rs\left( r+s\right) ^{2}}\dfrac{ \partial ^{2}f}{\partial y\partial x}-\dfrac{1}{\left( s+r\right) ^{2}} \dfrac{\partial f}{\partial y}+\dfrac{1}{\left( s+r\right) ^{2}}\dfrac{ \partial ^{2}f}{\partial x\partial y}+\dfrac{2s+r}{s\left( s+r\right) ^{2}} \dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = & -\dfrac{1}{\left( r+s\right) ^{2}}\dfrac{\partial f}{\partial x}+\dfrac{ 2r+s}{r\left( r+s\right) ^{2}}\dfrac{\partial ^{2}f}{\partial x^{2}}+\left( \dfrac{\left( 2r+s\right) \left( 2s+r\right) }{rs\left( r+s\right) ^{2}}+ \dfrac{1}{\left( s+r\right) ^{2}}\right) \dfrac{\partial ^{2}f}{\partial y\partial x}-\dfrac{1}{\left( s+r\right) ^{2}}\dfrac{\partial f}{\partial y}+ \dfrac{2s+r}{s\left( s+r\right) ^{2}}\dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}

Finalmente

\begin{eqnarray*} \dfrac{\partial ^{2}z}{\partial s^{2}} & = &\dfrac{\partial }{\partial s} \left( \dfrac{\partial z}{\partial s}\right) \\ & = &\dfrac{\partial }{\partial s}\left( \dfrac{1}{r+s}\dfrac{\partial f}{ \partial x}+\dfrac{2s+r}{s^{2}+rs}\dfrac{\partial f}{\partial y}\right) \\ & = &\dfrac{\partial }{\partial s}\left( \dfrac{1}{r+s}\right) \dfrac{\partial f}{\partial x}+\dfrac{1}{r+s}\dfrac{\partial }{\partial s}\left( \dfrac{ \partial f}{\partial x}\right) +\dfrac{\partial }{\partial s}\left( \dfrac{ 2s+r}{s^{2}+rs}\right) \dfrac{\partial f}{\partial y}+\dfrac{2s+r}{s^{2}+rs} \dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{\partial y}\right) \\ & = &-\dfrac{1}{\left( r+s\right) ^{2}}\dfrac{\partial f}{\partial x}+\dfrac{1 }{r+s}\left( \dfrac{\partial }{\partial x}\left( \dfrac{\partial f}{\partial x}\right) \dfrac{\partial x}{\partial s}+\dfrac{\partial }{\partial y}\left( \dfrac{\partial f}{\partial x}\right) \dfrac{\partial y}{\partial s}\right) + \dfrac{2\left( s^{2}+rs\right) -\left( 2s+r\right) \left( 2s+r\right) }{ \left( s^{2}+rs\right) ^{2}}\dfrac{\partial f}{\partial y}+\dfrac{2s+r}{ s^{2}+rs}\left( \dfrac{\partial }{\partial x}\left( \dfrac{\partial f}{ \partial y}\right) \dfrac{\partial x}{\partial s}+\dfrac{\partial }{\partial y}\left( \dfrac{\partial f}{\partial y}\right) \dfrac{\partial y}{\partial s} \right) \\ & = &-\dfrac{1}{\left( r+s\right) ^{2}}\dfrac{\partial f}{\partial x}+\dfrac{1 }{r+s}\left( \dfrac{\partial ^{2}f}{\partial x^{2}}\dfrac{\partial x}{ \partial s}+\dfrac{\partial ^{2}f}{\partial y\partial x}\dfrac{\partial y}{ \partial s}\right) +\dfrac{2\left( s^{2}+rs\right) -\left( 2s+r\right) \left( 2s+r\right) }{\left( s^{2}+rs\right) ^{2}}\dfrac{\partial f}{\partial y}+\dfrac{2s+r}{s^{2}+rs}\left( \dfrac{\partial ^{2}f}{\partial x\partial y} \dfrac{\partial x}{\partial s}+\dfrac{\partial ^{2}f}{\partial y^{2}}\dfrac{ \partial y}{\partial s}\right) \\ & = &-\dfrac{1}{\left( r+s\right) ^{2}}\dfrac{\partial f}{\partial x}+\dfrac{1 }{r+s}\left( \dfrac{\partial ^{2}f}{\partial x^{2}}\dfrac{1}{r+s}+\dfrac{ \partial ^{2}f}{\partial y\partial x}\dfrac{2s+r}{s^{2}+rs}\right) -\dfrac{ r^{2}+2rs+2s^{2}}{\left( s^{2}+rs\right) ^{2}}\dfrac{\partial f}{\partial y}+ \dfrac{2s+r}{s^{2}+rs}\left( \dfrac{\partial ^{2}f}{\partial x\partial y} \dfrac{1}{r+s}+\dfrac{\partial ^{2}f}{\partial y^{2}}\dfrac{2s+r}{s^{2}+rs} \right) \\ & = &-\dfrac{1}{\left( r+s\right) ^{2}}\dfrac{\partial f}{\partial x}+\dfrac{1 }{\left( r+s\right) ^{2}}\dfrac{\partial ^{2}f}{\partial x^{2}}+\dfrac{2s+r}{ \left( r+s\right) \left( s^{2}+rs\right) }\dfrac{\partial ^{2}f}{\partial y\partial x}-\dfrac{r^{2}+2rs+2s^{2}}{\left( s^{2}+rs\right) ^{2}}\dfrac{ \partial f}{\partial y}+\dfrac{2s+r}{\left( s^{2}+rs\right) \left( r+s\right) }\dfrac{\partial ^{2}f}{\partial x\partial y}+\dfrac{\left( 2s+r\right) ^{2}}{\left( s^{2}+rs\right) ^{2}}\dfrac{\partial ^{2}f}{ \partial y^{2}} \\ & = &-\dfrac{1}{\left( r+s\right) ^{2}}\dfrac{\partial f}{\partial x}+\dfrac{1 }{\left( r+s\right) ^{2}}\dfrac{\partial ^{2}f}{\partial x^{2}}+\dfrac{2s+r}{ s\left( r+s\right) ^{2}}\dfrac{\partial ^{2}f}{\partial y\partial x}-\dfrac{ r^{2}+2rs+2s^{2}}{\left( s^{2}+rs\right) ^{2}}\dfrac{\partial f}{\partial y}+ \dfrac{2s+r}{s\left( r+s\right) ^{2}}\dfrac{\partial ^{2}f}{\partial x\partial y}+\dfrac{\left( 2s+r\right) ^{2}}{\left( s^{2}+rs\right) ^{2}} \dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &-\dfrac{1}{\left( r+s\right) ^{2}}\dfrac{\partial f}{\partial x}+\dfrac{1 }{\left( r+s\right) ^{2}}\dfrac{\partial ^{2}f}{\partial x^{2}}+\dfrac{ 2\left( 2s+r\right) }{s\left( r+s\right) ^{2}}\dfrac{\partial ^{2}f}{ \partial y\partial x}-\dfrac{r^{2}+2rs+2s^{2}}{\left( s^{2}+rs\right) ^{2}} \dfrac{\partial f}{\partial y}+\dfrac{\left( 2s+r\right) ^{2}}{\left( s^{2}+rs\right) ^{2}}\dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}

En resumen

\begin{eqnarray*} \dfrac{\partial ^{2}z}{\partial r^{2}} & = & \dfrac{-2r^{2}-2rs-s^{2}}{\left( r^{2}+rs\right) ^{2}}\dfrac{\partial f}{\partial x}+\dfrac{\left( 2r+s\right) ^{2}}{\left( r^{2}+rs\right) ^{2}}\dfrac{\partial ^{2}f}{ \partial x^{2}}+\dfrac{2\left( 2r+s\right) }{\left( r^{2}+rs\right) \left( s+r\right) }\dfrac{\partial ^{2}f}{\partial y\partial x}-\dfrac{1}{\left( s+r\right) ^{2}}\dfrac{\partial f}{\partial y}+\dfrac{1}{\left( s+r\right) ^{2}}\dfrac{\partial ^{2}f}{\partial y^{2}} \\ \\ \dfrac{\partial ^{2}z}{\partial s\partial r} & = & -\dfrac{1}{\left( r+s\right) ^{2}}\dfrac{\partial f}{\partial x}+\dfrac{2r+s}{r\left( r+s\right) ^{2}} \dfrac{\partial ^{2}f}{\partial x^{2}}+\left( \dfrac{\left( 2r+s\right) \left( 2s+r\right) }{rs\left( r+s\right) ^{2}}+\dfrac{1}{\left( s+r\right) ^{2}}\right) \dfrac{\partial ^{2}f}{\partial y\partial x}-\dfrac{1}{\left( s+r\right) ^{2}}\dfrac{\partial f}{\partial y}+\dfrac{2s+r}{s\left( s+r\right) ^{2}}\dfrac{\partial ^{2}f}{\partial y^{2}} \\ \\ \dfrac{\partial ^{2}z}{\partial s^{2}} & = & -\dfrac{1}{\left( r+s\right) ^{2}} \dfrac{\partial f}{\partial x}+\dfrac{1}{\left( r+s\right) ^{2}}\dfrac{ \partial ^{2}f}{\partial x^{2}}+\dfrac{2\left( 2s+r\right) }{s\left( r+s\right) ^{2}}\dfrac{\partial ^{2}f}{\partial y\partial x}-\dfrac{ r^{2}+2rs+2s^{2}}{\left( s^{2}+rs\right) ^{2}}\dfrac{\partial f}{\partial y}+ \dfrac{\left( 2s+r\right) ^{2}}{\left( s^{2}+rs\right) ^{2}}\dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}