WebWork 14.6 Problem 5

Lots of people are asking about this problem. It states something along the lines of: If $z=\sin(x^2+y^2)$, $x=v\cos(u)$, and $y=v\sin(u)$, find $\partial z/\partial u$ and $\partial z/\partial v$. The variables are restricted to domains on which the functions are defined.

There is an easy way and a hard way to do this. Most of you are doing it the hard way, and your answer ends up being different than what WebWork is expecting.

First, notice that

\[z=\sin((v\cos(u))^2+(v\sin(u))^2)\]

and that this will simplify to a function of a single variable. Now, after doing the simplification, take your derivatives. If you try to take the derivatives too early, you get something that is technically correct, but far too complicated to be applicably correct.

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