Physical intuition behind inverse of derivative

Question

Suppose you do a transformation of variables from (x, a)-> (y, b) like from cartesian to polar.I know that $$\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}$$ This holds true for total derivatives but not for partial derivative. Does anybody has any logical explanation/intuition behind this?

Answer 1

Where did you get that it does not hold true for partial derivatives, which are just the usual single derivatives?

For instance, if you now let $y=f(z,w),$ and moreover if as a function of $z$ alone we have that $y$ is continuously differentiable and $y_z\ne 0,$ then the derivative of the inverse $y\mapsto z$ exists, and is still given by $$\frac{1}{\frac{\partial y}{\partial z}}.$$