I apologize about the title as I would have included the entire identity I am trying to prove analytically but we are limited to 150 characters.
I'm given the following problem in Widder's Advanced Calculus:
11.) $u=f\left(x,y\right),x=r\cos\left(\theta\right),y=r\sin\left(\theta\right).$ show that \begin{align} \left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2&=\left(\frac{\partial f}{\partial r}\right)^2+\frac{1}{r^2}\left(\frac{\partial f}{\partial\theta}\right)^2. \end{align}
This is what I have done so far: \begin{align} \left(\frac{\partial f\left(x,y\right)}{\partial x}\right)^2+\left(\frac{\partial f\left(x,y\right)}{\partial y}\right)^2&=\left(\frac{\partial f\left(r\cos\left(\theta\right),r\sin\left(\theta\right)\right)}{\partial\left(r\cos\left(\theta\right)\right)}\right)^2+\left(\frac{\partial f\left(r\cos\left(\theta\right),r\sin\left(\theta\right)\right)}{\partial \left(r\sin\left(\theta\right)\right)}\right)^2\\ \end{align} but is there a way to transform the RHS into what he has written? Perhaps via some substitution?