In Strauss PDE textbook, he derived the formula of Laplaican of spherical coordinate as follows:
changing $(x,y,z)$ to $(r,\theta,\phi)$: Use te notation $r = \sqrt{x^2+y^2+z^2} = \sqrt{s^2+z^2}$, $s = \sqrt{x^2+y^2}$, $x = s\cos\phi$, $y = s\sin\phi$, $z = r\cos\theta$, $s =r\sin\theta$. By the two-dimensional Laplace calculation, we have both $$u_{zz}+u_{ss} = u_{rr}+{1\over r}u_r+{1\over r^2}u_{\theta\theta}$$ and $$u_{xx}+u_{yy} = u_{ss} + {1\over s}u_{s}+{1\over s^2}u_{\phi,\phi}.$$ We add these two equations, and cancel $u_{ss}$, to get $$u_{xx}+u_{yy}+u_{zz} = u_{rr}+{1\over r}u_r+{1\over r^2}u_{\theta\theta}+{1\over s}u_s+{1\over s^2}u_{\phi\phi}.$$ In the last term we substitute $s^2 = r^2\sin^2\theta$ and in the next-to-last term \begin{align*} u_s & = {\partial u\over\partial s} = u_r{\partial r\over\partial s}+u_\theta{\partial\theta\over\partial s}+u_\phi{\partial\phi\over\partial s}\\ & = u_r{s\over r}+u_{\theta}{\cos\theta\over r}.\\ \end{align*}
I don't understand the last equality. Why ${\partial r\over\partial s} = {s\over r}$ and ${\partial\theta\over\partial s} = {\cos\theta\over r}$? From $s = r\sin\theta$, I can get $$1 = {\partial r\over\partial s}\sin\theta+r\cos\theta{\partial\theta\over\partial s}.$$
From the definitions $$r^2 = s^2 + z^2.$$ Implicit differentiation wrt $s$ gives $$ 2r \frac{\partial r}{\partial s} = 2s + 0$$ since $\partial z/\partial s = 0$, and this rearranges to the claim.