From a bank of exams:
Let $u(x,y) = f(r)$ be a smooth function in the plane that depends only on $r = \sqrt{x^2 + y^2}$. Compute $\Delta u = u_{xx} + u_{yy}$ in terms of $f$ and its derivatives.
Wikipedia states that the Laplace operator in polar coordinates is $$\Delta f = \frac{1}{r}\frac{\partial f}{\partial r} \left( r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2},$$ which I suppose I could memorize directly, but I thought there might be an easier way.
I tried to prove this directly, by thinking that $$ u_{xx} = \frac{d^2f}{dr^2} \frac{\partial r}{\partial x} + \frac{df}{dr} \frac{\partial ^2r}{\partial x^2}$$ and $$ u_{yy} = \frac{d^2f}{dr^2} \frac{\partial r}{\partial y} + \frac{df}{dr} \frac{\partial ^2r}{\partial y^2}.$$ But then I get stuck at $$ u_{xx} + u_{yy} = \frac{d^2f}{dr^2} \frac{x+y}{\sqrt{x^2+y^2}} + \frac{df}{dr}\frac{1}{\sqrt{x^2+y^2}} = \frac{d^2f}{dr^2} \frac{r(\cos \theta + \sin \theta)}{r} + \frac{df}{dr}\frac{1}{r}.$$ Any idea on where I'm going wrong? It looks like I need $\displaystyle{\frac{r(\cos \theta + \sin \theta)}{r} = 1}$.