Derivada Parcial

por **Pietro di Bernadone** Dom 02 Nov 2014, 20:21

Se z = f (x , y), onde x = r² + s² e y = 2rs, encontre $Derivada Parcial Gif$

Sei que a resolução sai pela regra da cadeia mas não confuso Derivada Parcial Icon_rolleyes

. Alguém me ajuda por favor?

por Man Utd Qua 05 Nov 2014, 20:47

$\\\\\\ \frac{\partial z }{\partial s}=\frac{\partial z }{\partial x}\frac{\partial x }{\partial s}+\frac{\partial z }{\partial y}\frac{\partial y }{\partial s} \\\\\\ \frac{\partial z }{\partial s}=2s\frac{\partial z }{\partial x}+2r\frac{\partial z }{\partial y}$

Derivando outra vez:

$\\\\\\ \frac{\partial }{\partial r} \left( \frac{\partial z }{\partial s} \right)=\frac{\partial }{\partial r}\left( 2s\frac{\partial z }{\partial x}+2r\frac{\partial z }{\partial y} \right) \\\\\\ \frac{\partial^{2} z }{\partial r\partial s} =\frac{\partial }{\partial r}\left( 2s\frac{\partial z }{\partial x} \right)+\frac{\partial }{\partial r}\left(2r\frac{\partial z }{\partial y} \right) \\\\\\ \frac{\partial^{2} z }{\partial r\partial s} = 2s\frac{\partial^{2} z }{\partial r\partial x} +2\frac{\partial z }{\partial y} +2r\frac{\partial^{2} z }{\partial r\partial y}$

Usando novamente a regra da cadeia:

$\\\\\\ \frac{\partial^{2} z }{\partial r\partial s} = 2s\left( \frac{\partial^{2} z }{\partial x^{2}} \frac{\partial x}{ \partial r} +\frac{\partial^{2} z }{\partial y \partial x} \frac{\partial y}{ \partial r} \right )+2\frac{\partial z }{\partial y} +2r\left( \frac{\partial^{2} z }{\partial y^{2}}\frac{\partial y}{ \partial r}+ \frac{\partial^{2} z }{\partial x \partial y}\frac{\partial x}{ \partial r} \right) \\\\\\ \frac{\partial^{2} z }{\partial r\partial s} = 2s\left( 2r \frac{\partial^{2} z }{\partial x^{2}} +2s\frac{\partial^{2} z }{\partial y \partial x} \right )+2\frac{\partial z }{\partial y} +2r\left( 2s\frac{\partial^{2} z }{\partial y^{2}}+ 2r\frac{\partial^{2} z }{\partial x \partial y} \right)$

$\\\\\\ \frac{\partial^{2} z }{\partial r\partial s} = 4rs \frac{\partial^{2} z }{\partial x^{2}} +4s^2\frac{\partial^{2} z }{\partial y \partial x} +2\frac{\partial z }{\partial y} +4rs\frac{\partial^{2} z }{\partial y^{2}}+ 4r^2\frac{\partial^{2} z }{\partial x \partial y}$

Lembre-se da propriedade da derivada mista : $\\\\\\ \frac{\partial^{2} z }{\partial x \partial y} =\frac{\partial^{2} z }{\partial y \partial x}$ logo :

$\\\\\\ \frac{\partial^{2} z }{\partial r\partial s} = 4rs \frac{\partial^{2} z }{\partial x^{2}} +4s^2\frac{\partial^{2} z }{\partial x \partial y} +2\frac{\partial z }{\partial y} +4rs\frac{\partial^{2} z }{\partial y^{2}}+ 4r^2\frac{\partial^{2} z }{\partial x \partial y} \\\\\\ \boxed{ \boxed{ \boxed{\frac{\partial^{2} z }{\partial r\partial s} = 4rs \frac{\partial^{2} z }{\partial x^{2}}+4rs\frac{\partial^{2} z }{\partial y^{2}} +(4s^2+4r^2)\frac{\partial^{2} z }{\partial x \partial y} +2\frac{\partial z }{\partial y} }}}$

Derivada Parcial

Derivada Parcial

Re: Derivada Parcial

Um fórum livre e democrático.