Partial Derivatives of Cartesian and Polar Coordinate Systems 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:


*

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

*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. 

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