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 \(\dfrac{\partial ^{2}z}{\partial t^{2}},\) \( \dfrac{\partial ^{2}z}{\partial s\partial t},\) \(\dfrac{\partial ^{2}z}{ \partial s^{2}} \) si \(x=e^{2t},\) \(y=e^{t-s}.\)

Solución:

Observamos el siguiente diagrama

La función \(f\) depende de \(x\) y \(y;\) \(x\) depende sólo de \(t\) y \(y\) dependen de \(t\) y \(s.\)

Calculamos

\begin{eqnarray*} \dfrac{\partial z}{\partial t} & = &\dfrac{\partial f}{\partial x}\dfrac{ \partial x}{\partial t}+\dfrac{\partial f}{\partial y}\dfrac{\partial y}{\partial t} \\ & = &\dfrac{\partial f}{\partial x}2e^{2t}+\dfrac{\partial f}{\partial y}e^{t-s} \\ & = &2e^{2t}\dfrac{\partial f}{\partial x}+e^{t-s}\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}0+\dfrac{\partial f}{\partial y}\left( -e^{t-s}\right) \\ & = &-e^{t-s}\dfrac{\partial f}{\partial y} \end{eqnarray*}

Ahora consideramos el siguiente diagrama

Calculamos

\begin{eqnarray*} \dfrac{\partial ^{2}z}{\partial t^{2}} & = &\dfrac{\partial }{\partial t} \left( \dfrac{\partial z}{\partial t}\right) \\ & = &\dfrac{\partial }{\partial t}\left( 2e^{2t}\dfrac{\partial f}{\partial x} +e^{t-s}\dfrac{\partial f}{\partial y}\right) \\ & = &\dfrac{\partial }{\partial t}\left( 2e^{2t}\right) \dfrac{\partial f}{ \partial x}+2e^{2t}\dfrac{\partial }{\partial t}\left( \dfrac{\partial f}{ \partial x}\right) +\dfrac{\partial }{\partial t}\left( e^{t-s}\right) \dfrac{\partial f}{\partial y}+e^{t-s}\dfrac{\partial }{\partial t}\left( \dfrac{\partial f}{\partial y}\right) \\ & = &4e^{2t}\dfrac{\partial f}{\partial x}+2e^{2t}\left( \dfrac{\partial }{ \partial x}\left( \dfrac{\partial f}{\partial x}\right) \dfrac{\partial x}{ \partial t}+\dfrac{\partial }{\partial y}\left( \dfrac{\partial f}{\partial x }\right) \dfrac{\partial y}{\partial t}\right) +e^{t-s}\dfrac{\partial f}{ \partial y}+e^{t-s}\left( \dfrac{\partial }{\partial x}\left( \dfrac{ \partial f}{\partial y}\right) \dfrac{\partial x}{\partial t}+\dfrac{ \partial }{\partial y}\left( \dfrac{\partial f}{\partial y}\right) \dfrac{ \partial y}{\partial t}\right) \\ & = &4e^{2t}\dfrac{\partial f}{\partial x}+2e^{2t}\left( \dfrac{\partial ^{2}f }{\partial x^{2}}\dfrac{\partial x}{\partial t}+\dfrac{\partial ^{2}f}{ \partial y\partial x}\dfrac{\partial y}{\partial t}\right) +e^{t-s}\dfrac{ \partial f}{\partial y}+e^{t-s}\left( \dfrac{\partial ^{2}f}{\partial x\partial y}\dfrac{\partial x}{\partial t}+\dfrac{\partial ^{2}f}{\partial y^{2}}\dfrac{\partial y}{\partial t}\right) \\ & = &4e^{2t}\dfrac{\partial f}{\partial x}+2e^{2t}\left( \dfrac{\partial ^{2}f }{\partial x^{2}}2e^{2t}+\dfrac{\partial ^{2}f}{\partial y\partial x} e^{t-s}\right) +e^{t-s}\dfrac{\partial f}{\partial y}+e^{t-s}\left( \dfrac{ \partial ^{2}f}{\partial x\partial y}2e^{2t}+\dfrac{\partial ^{2}f}{\partial y^{2}}e^{t-s}\right) \\ & = &4e^{2t}\dfrac{\partial f}{\partial x}+4e^{4t}\dfrac{\partial ^{2}f}{ \partial x^{2}}+2e^{2t}e^{t-s}\dfrac{\partial ^{2}f}{\partial y\partial x} +e^{t-s}\dfrac{\partial f}{\partial y}+2e^{2t}e^{t-s}\dfrac{\partial ^{2}f}{ \partial x\partial y}+e^{2t-2s}\dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &4e^{2t}\dfrac{\partial f}{\partial x}+4e^{4t}\dfrac{\partial ^{2}f}{ \partial x^{2}}+4e^{2t}e^{t-s}\dfrac{\partial ^{2}f}{\partial y\partial x} +e^{t-s}\dfrac{\partial f}{\partial y}+e^{2t-2s}\dfrac{\partial ^{2}f}{ \partial y^{2}} \end{eqnarray*}

La derivada parcial de segundo orden mixta es

\begin{eqnarray*} \dfrac{\partial ^{2}z}{\partial s\partial t} & = & \dfrac{\partial }{\partial s} \left( \dfrac{\partial z}{\partial t}\right) \\ & = & \dfrac{\partial }{\partial s}\left( 2e^{2t}\dfrac{\partial f}{\partial x} +e^{t-s}\dfrac{\partial f}{\partial y}\right) \\ &=&\dfrac{\partial }{\partial s}\left( 2e^{2t}\right) \dfrac{\partial f}{ \partial x}+2e^{2t}\dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{ \partial x}\right) +\dfrac{\partial }{\partial s}\left( e^{t-s}\right) \dfrac{\partial f}{\partial y}+e^{t-s}\dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{\partial y}\right) \\ & = & 0\dfrac{\partial f}{\partial x}+2e^{2t}\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) -e^{t-s}\dfrac{\partial f}{\partial y} +e^{t-s}\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) \\ & = & 2e^{2t}\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) -e^{t-s}\dfrac{\partial f}{\partial y}+e^{t-s}\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) \\ &=&2e^{2t}\left( \dfrac{\partial ^{2}f}{\partial x^{2}}0+\dfrac{\partial ^{2}f}{\partial y\partial x}\left( -e^{t-s}\right) \right) -e^{t-s}\dfrac{ \partial f}{\partial y}+e^{t-s}\left( \dfrac{\partial ^{2}f}{\partial x\partial y}0+\dfrac{\partial ^{2}f}{\partial y^{2}}\left( -e^{t-s}\right) \right) \\ & = & -2e^{2t}e^{t-s}\dfrac{\partial ^{2}f}{\partial y\partial x}-e^{t-s}\dfrac{ \partial f}{\partial y}-e^{2t-2s}\dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = & -2e^{3t-s}\dfrac{\partial ^{2}f}{\partial y\partial x}-e^{t-s}\dfrac{ \partial f}{\partial y}-e^{2t-2s}\dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}

y

\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( -e^{t-s}\dfrac{\partial f}{\partial y} \right) \\ & = &\dfrac{\partial }{\partial s}\left( -e^{t-s}\right) \dfrac{\partial f}{ \partial y}-e^{t-s}\dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{ \partial y}\right) \\ & = &e^{t-s}\dfrac{\partial f}{\partial y}-e^{t-s}\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) \\ & = &e^{t-s}\dfrac{\partial f}{\partial y}-e^{t-s}\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) \\ & = &e^{t-s}\dfrac{\partial f}{\partial y}-e^{t-s}\left( \dfrac{\partial ^{2}f }{\partial x\partial y}0+\dfrac{\partial ^{2}f}{\partial y^{2}}\left( -e^{t-s}\right) \right) \\ & = &e^{t-s}\dfrac{\partial f}{\partial y}+e^{2t-2s}\dfrac{\partial ^{2}f}{ \partial y^{2}} \end{eqnarray*}

En resumen

\begin{eqnarray*} \dfrac{\partial ^{2}z}{\partial t^{2}} & = & 4e^{2t}\dfrac{\partial f}{\partial x}+4e^{4t}\dfrac{\partial ^{2}f}{\partial x^{2}}+4e^{2t}e^{t-s}\dfrac{ \partial ^{2}f}{\partial y\partial x}+e^{t-s}\dfrac{\partial f}{\partial y} +e^{2t-2s}\dfrac{\partial ^{2}f}{\partial y^{2}} \\ \dfrac{\partial ^{2}z}{\partial s\partial t} & = & -2e^{3t-s}\dfrac{\partial ^{2}f}{\partial y\partial x}-e^{t-s}\dfrac{\partial f}{\partial y}-e^{2t-2s} \dfrac{\partial ^{2}f}{\partial y^{2}} \\ \dfrac{\partial ^{2}z}{\partial s^{2}} & = & e^{t-s}\dfrac{\partial f}{\partial y}+e^{2t-2s}\dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}