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:


`(\partial z)/(\partial u)=(\partial z)/(\partial x)(\partial x)/(\partial u)+(\partial z)/(\partial y)(\partial y)/(\partial u)`

`(\partial z)/(\partial u)=e^y(2u)+xe^y(2u)`

`(\partial z)/(\partial u)=2ue^y(1+x)`


`(\partial z)/(\partial v)=(\partial z)/(\partial x)(\partial x)/(\partial v)+(\partial z)/(\partial y)(\partial y)/(\partial v)`

`(\partial z)/(\partial v)=e^y(2v)+xe^y(-2v)`

`(\partial z)/(\partial v)=2ve^y(1-x)`