Quote:

= i z dx

integrating both sides.

[dz/z] = [i dx]

lnz = ix

From this point on I fully follow the proof however I don't think I understand the notation of dy/dx.

My current understanding is that when you differentiate a function y with respect to x you are calculating the gradient function and that is simply called (dy/dx). It could be called f'(x) or something else. However in this proof it's being treated like a fraction.

dy/dx is NOT fraction but it is a Quote:

If it is being treated like a fraction then what is dy, and what is dx? Isn't dx an infinitely small distance, when delta x has tended to 0?

Any help is much appreciated

dy/dx can be defined (and originally was by Newton and Leibniz) as a ratio of "infinitesmals". However, it turns out to be very difficult to give a rigorous definition of "infinitesmal" (Bishop Berkeley famously refered to infinitesmals as "ghosts of vanished quantities"!) and it was not until recently that that was done ("non-standard Analysis").