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 \(x=\tan \left( rs\right) ,\) \(y=\sec \left( rs\right) .\)

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( s\sec ^{2}\left( rs\right) \right) + \dfrac{\partial f}{\partial y}\left( s\sec \left( rs\right) \tan \left( rs\right) \right) \\ & = &s\sec ^{2}\left( rs\right) \dfrac{\partial f}{\partial x}+s\sec \left( rs\right) \tan \left( rs\right) \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( r\sec ^{2}\left( rs\right) \right) + \dfrac{\partial f}{\partial y}\left( r\sec \left( rs\right) \tan \left( rs\right) \right) \\ & = &r\sec ^{2}\left( rs\right) \dfrac{\partial f}{\partial x}+r\sec \left( rs\right) \tan \left( rs\right) \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( s\sec ^{2}\left( rs\right) \dfrac{ \partial f}{\partial x}+s\sec \left( rs\right) \tan \left( rs\right) \dfrac{ \partial f}{\partial y}\right) \\ & = &\dfrac{\partial }{\partial r}\left( s\sec ^{2}\left( rs\right) \right) \dfrac{\partial f}{\partial x}+s\sec ^{2}\left( rs\right) \dfrac{\partial }{ \partial r}\left( \dfrac{\partial f}{\partial x}\right) +\dfrac{\partial }{ \partial r}\left( s\sec \left( rs\right) \tan \left( rs\right) \right) \dfrac{\partial f}{\partial y}+ s\sec \left( rs\right) \tan \left( rs\right) \dfrac{\partial }{\partial r}\left( \dfrac{\partial f}{\partial y}\right) \\ & = &2s^{2}\sec ^{2}\left( rs\right) \tan \left( rs\right) \dfrac{\partial f}{ \partial x}+s\sec ^{2}\left( rs\right) \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) +\left( s^{2}\sec \left( rs\right) \tan ^{2}\left( rs\right) +s^{2}\sec ^{3}\left( rs\right) \right) \dfrac{ \partial f}{\partial y}+ \\ & &+ s\sec \left( rs\right) \tan \left( rs\right) \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) \\ & = &2s^{2}\sec ^{2}\left( rs\right) \tan \left( rs\right) \dfrac{\partial f}{ \partial x}+s\sec ^{2}\left( rs\right) \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) +\left( s^{2}\sec \left( rs\right) \tan ^{2}\left( rs\right) +s^{2}\sec ^{3}\left( rs\right) \right) \dfrac{\partial f}{\partial y}+ \\ & &+ s\sec \left( rs\right) \tan \left( rs\right) \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) \\ & = &2s^{2}\sec ^{2}\left( rs\right) \tan \left( rs\right) \dfrac{\partial f}{ \partial x}+s\sec ^{2}\left( rs\right) \left( \dfrac{\partial ^{2}f}{ \partial x^{2}}s\sec ^{2}\left( rs\right) +\dfrac{\partial ^{2}f}{\partial y\partial x}s\sec \left( rs\right) \tan \left( rs\right) \right) +\left( s^{2}\sec \left( rs\right) \tan ^{2}\left( rs\right) +s^{2}\sec ^{3}\left( rs\right) \right) \dfrac{\partial f}{\partial y} + \\ & &+s\sec \left( rs\right) \tan \left( rs\right) \left( \dfrac{\partial ^{2}f }{\partial x\partial y}s\sec ^{2}\left( rs\right) +\dfrac{\partial ^{2}f}{ \partial y^{2}}s\sec \left( rs\right) \tan \left( rs\right) \right) \\ & = &2s^{2}\sec ^{2}\left( rs\right) \tan \left( rs\right) \dfrac{\partial f}{ \partial x}+s^{2}\sec ^{4}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+s^{2}\sec ^{3}\left( rs\right) \tan \left( rs\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+\left( s^{2}\sec \left( rs\right) \tan ^{2}\left( rs\right) +s^{2}\sec ^{3}\left( rs\right) \right) \dfrac{\partial f}{\partial y}+ \\ & &+ s^{2}\sec ^{3}\left( rs\right) \tan \left( rs\right) \dfrac{\partial ^{2}f }{\partial x\partial y}+s^{2}\sec ^{2}\left( rs\right) \tan ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &2s^{2}\sec ^{2}\left( rs\right) \tan \left( rs\right) \dfrac{\partial f}{ \partial x}+s^{2}\sec ^{4}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+2s^{2}\sec ^{3}\left( rs\right) \tan \left( rs\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+\left( s^{2}\sec \left( rs\right) \tan ^{2}\left( rs\right) +s^{2}\sec ^{3}\left( rs\right) \right) \dfrac{\partial f}{\partial y}+ \\ & &+s^{2}\sec ^{2}\left( rs\right) \tan ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}

y

\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( s\sec ^{2}\left( rs\right) \dfrac{ \partial f}{\partial x}+s\sec \left( rs\right) \tan \left( rs\right) \dfrac{ \partial f}{\partial y}\right) \\ & = &\dfrac{\partial }{\partial s}\left( s\sec ^{2}\left( rs\right) \right) \dfrac{\partial f}{\partial x}+s\sec ^{2}\left( rs\right) \dfrac{\partial }{ \partial s}\left( \dfrac{\partial f}{\partial x}\right) +\dfrac{\partial }{ \partial s}\left( s\sec \left( rs\right) \tan \left( rs\right) \right) \dfrac{\partial f}{\partial y}+s\sec \left( rs\right) \tan \left( rs\right) \dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{\partial y}\right) \\ & = &\left( \sec ^{2}\left( rs\right) +2sr\sec ^{2}\left( rs\right) \tan \left( rs\right) \right) \dfrac{\partial f}{\partial x}+s\sec ^{2}\left( rs\right) \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) + \\ & &+ \left( \sec \left( rs\right) \tan \left( rs\right) +sr\tan ^{2}\left( rs\right) \sec \left( rs\right) +sr\sec ^{3}\left( rs\right) \right) \dfrac{ \partial f}{\partial y}+s\sec \left( rs\right) \tan \left( rs\right) \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) \\ & = &\left( \sec ^{2}\left( rs\right) +2sr\sec ^{2}\left( rs\right) \tan \left( rs\right) \right) \dfrac{\partial f}{\partial x}+s\sec ^{2}\left( rs\right) \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) + \\ & &+\left( \sec \left( rs\right) \tan \left( rs\right) +sr\tan ^{2}\left( rs\right) \sec \left( rs\right) +sr\sec ^{3}\left( rs\right) \right) \dfrac{ \partial f}{\partial y}+s\sec \left( rs\right) \tan \left( rs\right) \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) \\ & = &\left( \sec ^{2}\left( rs\right) +2sr\sec ^{2}\left( rs\right) \tan \left( rs\right) \right) \dfrac{\partial f}{\partial x}+s\sec ^{2}\left( rs\right) \left( \dfrac{\partial ^{2}f}{\partial x^{2}}r\sec ^{2}\left( rs\right) +\dfrac{\partial ^{2}f}{\partial y\partial x}r\sec \left( rs\right) \tan \left( rs\right) \right) + \\ & &+\left( \sec \left( rs\right) \tan \left( rs\right) +sr\tan ^{2}\left( rs\right) \sec \left( rs\right) +sr\sec ^{3}\left( rs\right) \right) \dfrac{ \partial f}{\partial y}+s\sec \left( rs\right) \tan \left( rs\right) \left( \dfrac{\partial ^{2}f}{\partial x\partial y}r\sec ^{2}\left( rs\right) + \dfrac{\partial ^{2}f}{\partial y^{2}}r\sec \left( rs\right) \tan \left( rs\right) \right) \\ & = &\left( \sec ^{2}\left( rs\right) +2sr\sec ^{2}\left( rs\right) \tan \left( rs\right) \right) \dfrac{\partial f}{\partial x}+s^{2}\sec ^{4}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+rs\sec ^{3}\left( rs\right) \tan \left( rs\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+ \\ & &+\left( \sec \left( rs\right) \tan \left( rs\right) +sr\tan ^{2}\left( rs\right) \sec \left( rs\right) +rs\sec ^{3}\left( rs\right) \right) \dfrac{ \partial f}{\partial y}+rs\sec ^{3}\left( rs\right) \tan \left( rs\right) \dfrac{\partial ^{2}f}{\partial x\partial y}+rs\sec ^{2}\left( rs\right) \tan ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &\left( \sec ^{2}\left( rs\right) +2sr\sec ^{2}\left( rs\right) \tan \left( rs\right) \right) \dfrac{\partial f}{\partial x}+s^{2}\sec ^{4}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+2rs\sec ^{3}\left( rs\right) \tan \left( rs\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+ \\ & &+\left( \sec \left( rs\right) \tan \left( rs\right) +sr\tan ^{2}\left( rs\right) \sec \left( rs\right) +rs\sec ^{3}\left( rs\right) \right) \dfrac{ \partial f}{\partial y}+rs\sec ^{2}\left( rs\right) \tan ^{2}\left( rs\right) \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( r\sec ^{2}\left( rs\right) \dfrac{ \partial f}{\partial x}+r\sec \left( rs\right) \tan \left( rs\right) \dfrac{ \partial f}{\partial y}\right) \\ & = &\dfrac{\partial }{\partial s}\left( r\sec ^{2}\left( rs\right) \right) \dfrac{\partial f}{\partial x}+r\sec ^{2}\left( rs\right) \dfrac{\partial }{ \partial s}\left( \dfrac{\partial f}{\partial x}\right) +\dfrac{\partial }{ \partial s}\left( r\sec \left( rs\right) \tan \left( rs\right) \right) \dfrac{\partial f}{\partial y}+r\sec \left( rs\right) \tan \left( rs\right) \dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{\partial y}\right) \\ & = &2r^{2}\sec ^{2}\left( rs\right) \tan \left( rs\right) \dfrac{\partial f}{ \partial x}+r\sec ^{2}\left( rs\right) \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) +\left( r^{2}\sec \left( rs\right) \tan ^{2}\left( rs\right) +r^{2}\sec ^{3}\left( rs\right) \right) \dfrac{ \partial f}{\partial y}+ \\ & &+r\sec \left( rs\right) \tan \left( rs\right) \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) \\ & = &2r^{2}\sec ^{2}\left( rs\right) \tan \left( rs\right) \dfrac{\partial f}{ \partial x}+r\sec ^{2}\left( rs\right) \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) +\left( r^{2}\sec \left( rs\right) \tan ^{2}\left( rs\right) +r^{2}\sec ^{3}\left( rs\right) \right) \dfrac{\partial f}{\partial y}+ \\ & &+r\sec \left( rs\right) \tan \left( rs\right) \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) \\ & = &2r^{2}\sec ^{2}\left( rs\right) \tan \left( rs\right) \dfrac{\partial f}{ \partial x}+r\sec ^{2}\left( rs\right) \left( \dfrac{\partial ^{2}f}{ \partial x^{2}}r\sec ^{2}\left( rs\right) +\dfrac{\partial ^{2}f}{\partial y\partial x}r\sec \left( rs\right) \tan \left( rs\right) \right) +\left( r^{2}\sec \left( rs\right) \tan ^{2}\left( rs\right) +r^{2}\sec ^{3}\left( rs\right) \right) \dfrac{\partial f}{\partial y}+ \\ & &+r\sec \left( rs\right) \tan \left( rs\right) \left( \dfrac{\partial ^{2}f }{\partial x\partial y}r\sec ^{2}\left( rs\right) +\dfrac{\partial ^{2}f}{ \partial y^{2}}r\sec \left( rs\right) \tan \left( rs\right) \right) \\ & = &2r^{2}\sec ^{2}\left( rs\right) \tan \left( rs\right) \dfrac{\partial f}{ \partial x}+r^{2}\sec ^{4}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+r^{2}\sec ^{3}\left( rs\right) \tan \left( rs\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+\left( r^{2}\sec \left( rs\right) \tan ^{2}\left( rs\right) +r^{2}\sec ^{3}\left( rs\right) \right) \dfrac{\partial f}{\partial y}+ \\ & &+r^{2}\sec ^{3}\left( rs\right) \tan \left( rs\right) \dfrac{\partial ^{2}f }{\partial x\partial y}+r^{2}\sec ^{2}\left( rs\right) \tan ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &2r^{2}\sec ^{2}\left( rs\right) \tan \left( rs\right) \dfrac{\partial f}{ \partial x}+r^{2}\sec ^{4}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+2r^{2}\sec ^{3}\left( rs\right) \tan \left( rs\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+\left( r^{2}\sec \left( rs\right) \tan ^{2}\left( rs\right) +r^{2}\sec ^{3}\left( rs\right) \right) \dfrac{\partial f}{\partial y}+ \\ & &+r^{2}\sec ^{2}\left( rs\right) \tan ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}

En resumen

\begin{eqnarray*} \dfrac{\partial ^{2}z}{\partial r^{2}} & = &2s^{2}\sec ^{2}\left( rs\right) \tan \left( rs\right) \dfrac{\partial f}{\partial x}+s^{2}\sec ^{4}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+2s^{2}\sec ^{3}\left( rs\right) \tan \left( rs\right) \dfrac{\partial ^{2}f}{\partial y\partial x} +\left( s^{2}\sec \left( rs\right) \tan ^{2}\left( rs\right) +s^{2}\sec ^{3}\left( rs\right) \right) \dfrac{\partial f}{\partial y}+ \\ & &+ s^{2}\sec ^{2}\left( rs\right) \tan ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{ \partial y^{2}} \\ \dfrac{\partial ^{2}z}{\partial s\partial r} & = &\left( \sec ^{2}\left( rs\right) +2sr\sec ^{2}\left( rs\right) \tan \left( rs\right) \right) \dfrac{ \partial f}{\partial x}+s^{2}\sec ^{4}\left( rs\right) \dfrac{\partial ^{2}f }{\partial x^{2}}+2rs\sec ^{3}\left( rs\right) \tan \left( rs\right) \dfrac{ \partial ^{2}f}{\partial y\partial x}+ \\ & &+ \left( \sec \left( rs\right) \tan \left( rs\right) +sr\tan ^{2}\left( rs\right) \sec \left( rs\right) + rs\sec^{3}\left( rs\right) \right) \dfrac{\partial f}{\partial y}+rs\sec ^{2}\left( rs\right) \tan ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{ \partial y^{2}} \\ \dfrac{\partial ^{2}z}{\partial s^{2}} & = &2r^{2}\sec ^{2}\left( rs\right) \tan \left( rs\right) \dfrac{\partial f}{\partial x}+r^{2}\sec ^{4}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+2r^{2}\sec ^{3}\left( rs\right) \tan \left( rs\right) \dfrac{\partial ^{2}f}{\partial y\partial x} +\left( r^{2}\sec \left( rs\right) \tan ^{2}\left( rs\right) +r^{2}\sec ^{3}\left( rs\right) \right) \dfrac{\partial f}{\partial y}+ \\ & &+ r^{2}\sec^{2}\left( rs\right) \tan ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{ \partial y^{2}} \end{eqnarray*}