[Math] Partial Derivatives of Cartesian and Polar Coordinate Systems

Question

Plane polar coordinates, $r$ and $\phi$, and Cartesian coordinates, $x$ and $y$, are related by
$$x=r\cos\phi,\ y=r\sin\phi,$$
and
$$r^2=x^2+y^2,\ \phi=\tan^{-1}({y \over x}).$$

To try to find the partial derivative $\partial r \over \partial x$, I used two approaches:

1. Using $x=r \cos \phi$,
   $${\partial x \over \partial r}=\cos \phi,$$
   and using ${\partial x \over \partial r}=({\partial r \over \partial x})^{-1},$
   $${\partial r \over \partial x}=\frac{1}{\cos \phi}.$$
2. Using $r^2=x^2+y^2$,
   $$r=(x^2+y^2)^{1/2},$$
   $$\frac{\partial r}{\partial x}=\frac{x}{(x^2+y^2)^{1/2}}=\cos\phi.$$

Why are the two answers different? I think the second approach gives the correct answer but I do not understand what is wrong with the first approach.

Answer 1

Division in partial derivatives is just a notation so $\frac{\partial r}{\partial x} \neq \frac{\partial x}{\partial r}^{-1}$.