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

Ejemplos de la regla de la cadena

En la siguiente lista \(z=f\left( x,y\right) \) es una función real de clase $C^{2}$ en $\mathbb{R}^{2}.$

Ejemplo 1 Calcula \(\dfrac{\partial ^{2}z}{\partial r^{2}},\) \(\dfrac{\partial ^{2}z}{\partial s\partial r},\) \(\dfrac{\partial ^{2}z}{\partial s^{2}}\) si \(x=r^{4}+s^{2},\) \(y=r^{3}+s^{3}.\)

Ejemplo 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 \(x=r^{2}s^{3},\) \(y=rs^{4}.\)

Ejemplo 3 Calcula \(\dfrac{\partial ^{2}z}{\partial r^{2}},\) \(\dfrac{\partial ^{2}z}{\partial s\partial r},\) \(\dfrac{\partial ^{2}z}{\partial s^{2}}\) si \(x=r^{4}+s^{2},\) \(y=5r^{3}s^{2}.\)

Ejemplo 4 Calcula \(\dfrac{\partial ^{2}z}{\partial r^{2}},\) \(\dfrac{\partial ^{2}z}{\partial s\partial r},\) \(\dfrac{\partial ^{2}z}{\partial s^{2}}\) si \(x=\dfrac{r}{s},\) \(y=rs.\)

Ejemplo 5 Calcula \(\dfrac{\partial ^{2}z}{\partial r^{2}},\) \(\dfrac{\partial ^{2}z}{\partial \theta \partial r},\) \(\dfrac{\partial ^{2}z}{\partial \theta^{2}}\) si \(x=r\cos \theta ,\) \(y=r\ \text{sen} \ \theta .\)

Ejemplo 6 Calcula \(\dfrac{\partial ^{2}z}{\partial r^{2}},\) \(\dfrac{\partial ^{2}z}{\partial s\partial r},\) \(\dfrac{\partial ^{2}z}{\partial s^{2}}\) si \(x=\cos \left( rs\right) ,\) \(y=\ \text{sen}\ \left( rs\right) .\)

Ejemplo 7 Calcula \(\dfrac{\partial ^{2}z}{\partial r^{2}},\) \(\dfrac{\partial ^{2}z}{\partial s\partial r},\) \(\dfrac{\partial ^{2}z}{\partial s^{2}}\) si \(x=\ \text{sen}\ \left( 2s+r\right) ,\) \(y=\cos \left( 2s-2r\right) .\)

Ejemplo 8 Calcula \(\dfrac{\partial ^{2}z}{\partial r^{2}},\) \(\dfrac{\partial ^{2}z}{\partial s\partial r},\) \(\dfrac{\partial ^{2}z}{\partial s^{2}}\) si \(x=\tan \dfrac{r}{s},\) \(y=\cot \dfrac{s}{r}.\)

Ejemplo 9 Calcula \(\dfrac{\partial ^{2}z}{\partial r^{2}},\) \(\dfrac{\partial ^{2}z}{\partial s\partial r},\) \(\dfrac{\partial ^{2}z}{\partial s^{2}}\) si \(x=\tan r^{2}s,\) \(y=\sec rs^{2}.\)

Ejemplo 10 Calcula \(\dfrac{\partial ^{2}z}{\partial r^{2}},\) \(\dfrac{\partial ^{2}z}{\partial s\partial r},\) \(\dfrac{\partial ^{2}z}{\partial s^{2}}\) si \(x=r^{3}\ \text{sen}\ 2s,\) \(y=r^{2}\cos 3s.\)

Ejemplo 11 Calcula \(\dfrac{\partial ^{2}z}{\partial r^{2}},\) \(\dfrac{\partial ^{2}z}{\partial s\partial r},\) \(\dfrac{\partial ^{2}z}{\partial s^{2}}\) si \(x=e^{2t},\) \(y=e^{t-s}.\)

Ejemplo 12 Calcula \(\dfrac{\partial ^{2}z}{\partial r^{2}},\) \(\dfrac{\partial ^{2}z}{\partial s\partial r},\) \(\dfrac{\partial ^{2}z}{\partial s^{2}}\) si \(x=\ln \left( r^{2}+rs\right) ,\) \(y=\ln \left( s^{2}+rs\right) .\)