I'm reading the derivation in Strauss's PDE book. He sets up the following relations:
$$ r = \sqrt{x^2 + y^2 +z^2} = \sqrt{s^2 + z^2} , s = \sqrt{x^2 + y^2}, x = s \cos{\phi}, y - s \sin{\phi}, z = r \cos{\theta}, s = r \sin{\theta} $$.
At one point, Strauss computes $u_{s}$(where $u$ is the function which satisfies Laplace's equation).
$$ u_{s} = u_{r} \frac{\partial r}{\partial s}+ u_{\theta}\frac{\partial \theta}{ \partial s} + u_{\phi} \frac{\partial \phi}{\partial s} $$
$$ u_{r} . \frac{s}{r} + u_{\theta}.\frac{cos {\theta}}{r} + 0 $$
I don't understand how he got that expression for $ \frac{\partial \theta}{ \partial s} $. Since, $ s= r \sin{\theta} $, it follows that : $$ r \cos{\theta} \frac{\partial \theta}{\partial s} = 1 $$ and hence: $$ \frac{\partial \theta}{\partial s} = \frac{1}{r \cos{\theta}} $$
Where am I making an error? Is my use of implicit differentiation incorrect?
Hint:
In the derivative of $s=r\sin \theta$ you have forgot the derivative of $r$:
the correct result is $$ 1=\frac{s}{r}\sin \theta+r\cos \theta \frac{\partial \theta}{\partial s} $$