If $ u = f(x,y)$, where $x=e^s \cos t$ and $y = e^s \sin t$, show that $ \frac{\partial ^2u}{\partial x^2} + \frac{\partial ^2u}{\partial y^2}= e^{-2s}\left[\frac{\partial ^2 u}{\partial s^2}+ \frac{\partial ^2 u}{\partial t^2}\right].$
Currently, what I've done:
$\begin{align} f_x &= (f_s, f_t), \\f_s &= e^s \cos t, \\f_t &= -e^s \sin t, \\(f_x)_x &= ((f_s)_s, (f_t)_t), \\(f_s)_s &= e^s \cos t, \\(f_s)_t &= -e^s \sin t \\(f_t)_s &= -e^s \sin t \\ (f_t)_t &= -e^s \cos t \\ \ \\f_y &=(f_s, f_t), \\f_s &= e^s \sin t, \\f_t &= e^s \cos t, \\(f_y)_y &= ((f_s)_s, (f_t)_t), \\(f_s)_s &= e^s \sin t, \\(f_s)_t &= e^s \cos t \\(f_t)_s &= e^s \cos t \\ (f_t)_t &= -e^s \sin t \end{align}$
Could tutors over here advise me whether I am on the right track, as I have no idea how to proceed from here. Thanks :)