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I have to find $dz/dx$ and $dz/dy$ if $z=u^3-uv+v^2$ ,$u=x/y$ and $v=xy$...Maybe I should express them in a different way so I can integrate them,but how?

ABC
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Tweetingbird
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  • see http://math.stackexchange.com/questions/348811/how-to-find-fracdzdx-and-fracdzdy-if-z-lnexey-and-y-x3/348823#348823 You have to find relations in du,dx,dy using the given relations and then use chain rule while differentiating z – ABC Apr 02 '13 at 07:26

2 Answers2

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I believe in this case

$$\frac{dz}{dx}=\frac{\partial z}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial z}{\partial v}\frac{\partial v}{\partial x}$$

and similarly for $\frac{dz}{dy}$.

Mike
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$u=x/y$ => $du=\dfrac{ydx-xdy}{y^2}$

and $v=xy$ => $dv=xdy+ydx$

$dz=3u^2du-vdu-udv+2vdv$

Put values of $du$ and $dv$ and divide by $dx$ and $dy$ to get $dz/dx$ and $dz/dy$ respectively.

ABC
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