In case of an implicit function $f(x,y)=0$, $ \frac{dy}{dx}= \frac{1}{\frac{dx}{dy}}$ , but if $f(x,y)=0 $ such that $ x=g(r,\theta) $ and $y=h(r,\theta)$ then why $\frac{\partial x}{\partial r} \ne\frac{1}{\frac{\partial r}{\partial x}} $ ?
I am a beginner and found it intriguing while converting the coordinates in my mechanical engineering course.
When you compute $\partial x/\partial r$, it's understood that you're supposed to keep $\theta$ constant, but when you compute $\partial r/\partial y$, you're supposed to keep $x$ constant, which is something completely different. In view of this, there shouldn't be any reason to expect that $\partial x/\partial r = 1/(\partial r/\partial x)$.
What is true, however, is that the Jacobian matrices are each other's inverses: $$ \begin{pmatrix} \partial x/\partial r & \partial x/\partial \theta \\ \partial y/\partial r & \partial y/\partial \theta \end{pmatrix} = \begin{pmatrix} \partial r/\partial x & \partial r/\partial y \\ \partial \theta/\partial x & \partial \theta/\partial y \end{pmatrix}^{-1} . $$