Section 14.5 Chain Rule

Assume that all the given functions are differentiable. If \(u = f(x, y)\), where \(x = e^s \cos t\) and \(y = e^s \sin t\), show that \[ \left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial u}{\partial y}\right)^2 = e^{-2s}\left[\left(\frac{\partial u}{\partial s}\right)^2 + \left(\frac{\partial u}{\partial t}\right)^2\right] \]