Suppose that at the point (x,y) y=f(x) is differentiable and the derivative is non-zero.

Plot the curve, also plot its reflection about the line y=x. Now consider the slope at (x,y) on the original curve with that on the reflected curve at (y,x) (the points image under reflection).

Now make the idea fit your course's idea of rigour.

PS you cannot prove anything with Lebnitz notation its just a shorthand for the limit defintion of derivative, you will have to go back to that definition or use a geometric argument like that above.

CB