In polar coordinates there is the following definition: $ x = \rho \cos \phi $, from which it follows that $ \frac{ \mathrm d x}{\mathrm d \rho} = \cos \phi $ and that $ \rho = x \cos^{-1} \phi $. Also $ \frac{ \mathrm d \rho}{\mathrm d x} = \cos^{-1} \phi $. On the other hand, there is the definition $ \rho = \sqrt {x^2 + y^2} $, from which it follows that $$ \frac{\mathrm d \rho}{\mathrm d x} = \frac{1}{2} \left( x^2 + y^2 \right)^{-1/2} \,2 x = \frac{x}{\sqrt {x^2 + y^2}} = \cos \phi $$.
These results make no sense: $\cos \phi$ is not generally equal to $\cos ^ {-1} \phi$. If somebody would indicate the error of my logic then I say thank you.
The error is the implicit assumptions you are making. In the first case, when you took a partial derivative of $\rho = x\cos^{-1}\phi$ wrt $x$, you assumed $\phi$ is not changing, i.e., the derivative is along a line going to or from the origin (i.e., $y$ is changing in such a way that $\phi$ remains constant). However, when you took the other partial derivative of $\rho = \sqrt{x^2 + y^2}$ wrt $x$, you assume $y$ is not changing, i.e., the derivative is along a line going up or down vertically (i.e., $\phi$ is changing in such as way that $y$ remains constant). There is no reason to expect the $2$ results to be the same in general, as you have noticed yourself.