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 \(x=\ \text{sen}\ \left( 2s+r\right) ,\) \(y=\cos \left( 2s-2r\right) .\)

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( \cos \left( 2s+r\right) \right) + \dfrac{\partial f}{\partial y}\left( 2\ \text{sen}\ \left( 2s-2r\right) \right) \\ & = &\cos \left( 2s+r\right) \dfrac{\partial f}{\partial x}+2\ \text{sen}\ \left( 2s-2r\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( 2\cos \left( 2s+r \right) \right)+ \dfrac{\partial f}{\partial y}\left( -2\ \text{sen}\ \left( 2s-2r\right) \right) \\ & = &2\cos \left( 2s+r\right) \dfrac{\partial f}{\partial x}-2\ \text{sen}\ \left( 2s-2r\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( \cos \left( 2s+r\right) \dfrac{ \partial f}{\partial x}+2\ \text{sen}\ \left( 2s-2r\right) \dfrac{\partial f}{ \partial y}\right) \\ & = &\dfrac{\partial }{\partial r}\left( \cos \left( 2s+r\right) \right) \dfrac{\partial f}{\partial x}+\cos \left( 2s+r\right) \dfrac{\partial }{ \partial r}\left( \dfrac{\partial f}{\partial x}\right) +\dfrac{\partial }{ \partial r}\left( 2\ \text{sen}\ \left( 2s-2r\right) \right) \dfrac{\partial f }{\partial y}+2\ \text{sen}\ \left( 2s-2r\right) \dfrac{\partial }{\partial r} \left( \dfrac{\partial f}{\partial y}\right) \\ & = &-\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+\cos \left( 2s+r\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) -4\cos \left( 2s-2r\right) \dfrac{\partial f}{ \partial y}+ \\ & &+2\ \text{sen}\ \left( 2s-2r\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) \\ & = &-\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+\cos \left( 2s+r\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) -4\cos \left( 2s-2r\right) \dfrac{\partial f}{ \partial y}+2\ \text{sen}\ \left( 2s-2r\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) \\ & = &-\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+\cos \left( 2s+r\right) \left( \dfrac{\partial ^{2}f}{\partial x^{2}}\cos \left( 2s+r\right) +\dfrac{\partial ^{2}f}{\partial y\partial x}\left( 2\ \text{sen}\ \left( 2s-2r\right) \right) \right) -4\cos \left( 2s-2r\right) \dfrac{ \partial f}{\partial y}+ \\ & &+2\ \text{sen}\ \left( 2s-2r\right) \left( \dfrac{ \partial ^{2}f}{\partial x\partial y}\cos \left( 2s+r\right) +\dfrac{ \partial ^{2}f}{\partial y^{2}}\left( 2\ \text{sen}\ \left( 2s-2r\right) \right) \right) \\ & = &-\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+\cos ^{2}\left( 2s+r\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+2\cos \left( 2s+r\right) \ \text{sen}\ \left( 2s-2r\right) \dfrac{\partial ^{2}f}{\partial y\partial x}-4\cos \left( 2s-2r\right) \dfrac{\partial f}{\partial y}+ \\ & &+2\ \text{sen}\ \left( 2s-2r\right) \cos \left( 2s+r\right) \dfrac{\partial ^{2}f}{\partial x\partial y}+4\ \text{sen} ^{2}\left( 2s-2r\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &-\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+\cos ^{2}\left( 2s+r\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+4\cos \left( 2s+r\right) \ \text{sen}\ \left( 2s-2r\right) \dfrac{\partial ^{2}f}{\partial y\partial x}-4\cos \left( 2s-2r\right) \dfrac{\partial f}{\partial y}+4 \ \text{sen} ^{2}\left( 2s-2r\right) \dfrac{\partial ^{2}f}{ \partial y^{2}} \end{eqnarray*}

La derivada 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( \cos \left( 2s+r\right) \dfrac{ \partial f}{\partial x}+2\ \text{sen}\ \left( 2s-2r\right) \dfrac{\partial f}{ \partial y}\right) \\ & = &\dfrac{\partial }{\partial s}\left( \cos \left( 2s+r\right) \right) \dfrac{\partial f}{\partial x}+\cos \left( 2s+r\right) \dfrac{\partial }{ \partial s}\left( \dfrac{\partial f}{\partial x}\right) +\dfrac{\partial }{ \partial s}\left( 2\ \text{sen}\ \left( 2s-2r\right) \right) \dfrac{\partial f }{\partial y}+2\ \text{sen}\ \left( 2s-2r\right) \dfrac{\partial }{\partial s} \left( \dfrac{\partial f}{\partial y}\right) \\ & = &-2\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+\cos \left( 2s+r\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( 4\cos \left( 2s-2r\right) \right) \dfrac{\partial f}{\partial y}+ \\ & &+2\ \text{sen}\ \left( 2s-2r\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) \\ & = &-2\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+\cos \left( 2s+r\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( 4\cos \left( 2s-2r\right) \right) \dfrac{\partial f}{\partial y}+2\ \text{sen}\ \left( 2s-2r\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) \\ & = &-2\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+\cos \left( 2s+r\right) \left( \dfrac{\partial ^{2}f}{\partial x^{2}}\left( 2\cos \left( 2s+r\right) \right) +\dfrac{\partial ^{2}f}{\partial y\partial x} \left( -2\ \text{sen}\ \left( 2s-2r\right) \right) \right) +4\cos \left( 2s-2r\right) \dfrac{\partial f}{\partial y}+ \\ & &+2\ \text{sen}\ \left( 2s-2r\right) \left( \dfrac{\partial ^{2}f}{\partial x\partial y}\left( 2\cos \left( 2s+r\right) \right) +\dfrac{\partial ^{2}f}{\partial y^{2}}\left( -2 \ \text{sen}\ \left( 2s-2r\right) \right) \right) \\ & = &-2\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+2\cos ^{2}\left( 2s+r\right) \dfrac{\partial ^{2}f}{\partial x^{2}}-2\cos \left( 2s+r\right) \ \text{sen}\ \left( 2s-2r\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+4\cos \left( 2s-2r\right) \dfrac{\partial f}{\partial y}+ \\ & &+4 \ \text{sen}\ \left( 2s-2r\right) \cos \left( 2s+r\right) \dfrac{\partial ^{2}f}{\partial x\partial y}-4\ \text{sen} ^{2}\left( 2s-2r\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &-2\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+2\cos ^{2}\left( 2s+r\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+2\cos \left( 2s+r\right) \ \text{sen}\ \left( 2s-2r\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+4\cos \left( 2s-2r\right) \dfrac{\partial f}{\partial y}-4 \ \text{sen} ^{2}\left( 2s-2r\right) \dfrac{\partial ^{2}f}{ \partial y^{2}} \end{eqnarray*}

Ahora calculamos la otra derivada parcial de segundo orden

\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( 2\cos \left( 2s+r\right) \dfrac{ \partial f}{\partial x}-2\ \text{sen}\ \left( 2s-2r\right) \dfrac{\partial f}{ \partial y}\right) \\ & = &\dfrac{\partial }{\partial s}\left( 2\cos \left( 2s+r\right) \right) \dfrac{\partial f}{\partial x}+2\cos \left( 2s+r\right) \dfrac{\partial }{ \partial s}\left( \dfrac{\partial f}{\partial x}\right) +\dfrac{\partial }{ \partial s}\left( -2\ \text{sen}\ \left( 2s-2r\right) \right) \dfrac{\partial f}{\partial y}-2\ \text{sen}\ \left( 2s-2r\right) \dfrac{\partial }{\partial s }\left( \dfrac{\partial f}{\partial y}\right) \\ & = &-4\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+2\cos \left( 2s+r\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) -4\cos \left( 2s-2r\right) \dfrac{\partial f}{ \partial y}+ \\ & &-2\ \text{sen}\ \left( 2s-2r\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) \\ & = &-4\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+2\cos \left( 2s+r\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) -4\cos \left( 2s-2r\right) \dfrac{\partial f}{ \partial y} + \\ & &-2\ \text{sen}\ \left( 2s-2r\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) \\ & = &-4\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+2\cos \left( 2s+r\right) \left( \dfrac{\partial ^{2}f}{\partial x^{2}}\left( 2\cos \left( 2s+r\right) \right) +\dfrac{\partial ^{2}f}{\partial y\partial x} \left( -2\ \text{sen}\ \left( 2s-2r\right) \right) \right) -4\cos \left( 2s-2r\right) \dfrac{\partial f}{\partial y}+\\ & &-2\ \text{sen}\ \left( 2s-2r\right) \left( \dfrac{\partial ^{2}f}{\partial x\partial y}\left( 2\cos \left( 2s+r\right) \right) +\dfrac{\partial ^{2}f}{\partial y^{2}}\left( -2 \ \text{sen}\ \left( 2s-2r\right) \right) \right) \\ & = &-4\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+4\cos ^{2}\left( 2s+r\right) \dfrac{\partial ^{2}f}{\partial x^{2}}-4\cos \left( 2s+r\right) \ \text{sen}\ \left( 2s-2r\right) \dfrac{\partial ^{2}f}{\partial y\partial x}-4\cos \left( 2s-2r\right) \dfrac{\partial f}{\partial y}+ \\ & &-4 \ \text{sen}\ \left( 2s-2r\right) \cos \left( 2s+r\right) \dfrac{\partial ^{2}f}{\partial x\partial y}+4\ \text{sen} ^{2}\left( 2s-2r\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &-4\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+4\cos ^{2}\left( 2s+r\right) \dfrac{\partial ^{2}f}{\partial x^{2}}-8\cos \left( 2s+r\right) \ \text{sen}\ \left( 2s-2r\right) \dfrac{\partial ^{2}f}{\partial y\partial x}-4\cos \left( 2s-2r\right) \dfrac{\partial f}{\partial y}+4 \ \text{sen} ^{2}\left( 2s-2r\right) \dfrac{\partial ^{2}f}{ \partial y^{2}}. \end{eqnarray*}

Por lo tanto,

\begin{eqnarray*} \dfrac{\partial ^{2}z}{\partial r^{2}} & = &-\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+\cos ^{2}\left( 2s+r\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+4\cos \left( 2s+r\right) \ \text{sen}\ \left( 2s-2r\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+4\cos \left( 2s-2r\right) \dfrac{\partial f}{\partial y}+4\ \text{sen} ^{2}\left( 2s-2r\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ \dfrac{\partial ^{2}z}{\partial s\partial r} & = &2\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+2\cos ^{2}\left( 2s+r\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+2\cos \left( 2s+r\right) \ \text{sen}\ \left( 2s-2r\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+4\cos \left( 2s-2r\right) \dfrac{\partial f}{\partial y}-4\ \text{sen} ^{2}\left( 2s-2r\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ \dfrac{\partial ^{2}z}{\partial s^{2}} & = &-4\ \text{sen}\ \left( 2s+r\right) \dfrac{\partial f}{\partial x}+4\cos ^{2}\left( 2s+r\right) \dfrac{\partial ^{2}f}{\partial x^{2}}-8\cos \left( 2s+r\right) \ \text{sen}\ \left( 2s-2r\right) \dfrac{\partial ^{2}f}{\partial y\partial x}-4\cos \left( 2s-2r\right) \dfrac{\partial f}{\partial y}+4\ \text{sen} ^{2}\left( 2s-2r\right) \dfrac{\partial ^{2}f}{\partial y^{2}}. \end{eqnarray*}