I was hoping someone might be able to clear up a fundamental issue that I am having trying to follow a proof for the Euler Formula, as follows.

(Sorry I'm not sure how to make the equations look pretty)

z=cosx + i sinx

dz = i(cosx + i sinx) dx

= 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.

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