Question

If z=xe^y, x=u^2+v^2, y=u^2-v^2, find dz/du and dz/dv. these variables are restrticted to domains on which the dunctions are defined

Answer 1

Chain rule:

$\dfrac{\partial z}{\partial u}=\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial u}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial u}$
$\dfrac{\partial z}{\partial u}=e^y(2u)+xe^y(2u)$
$\dfrac{\partial z}{\partial u}=2ue^y(1+x)$

$\dfrac{\partial z}{\partial v}=\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial v}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial v}$
$\dfrac{\partial z}{\partial v}=e^y(2v)+xe^y(-2v)$
$\dfrac{\partial z}{\partial v}=2ve^y(1-x)$