calculus - Notation of derivatives...

I asked my teacher the difference between this notations.

(1) $$\frac{dy}{dx}$$

(2) $$\frac{\delta y}{\delta x}$$
(3) $$\frac{\Delta y}{\Delta x}$$

He told me that there is no difference.

I really don't think he's right...

Question:

I think that (1) and (2) is more like the convention expressing the limit of a fraction. (3) instead really represent de ratio of the increments of y and x

Am I right?

Answer

Typically,

1. $\displaystyle\frac{dy}{dx}$ is the derivative (the slope of the tangent line);




2. $\displaystyle\frac{\delta y}{\delta x(t)}$ is a functional derivative where $y=y[x]$ is a functional of $x(t)$;




3. $\displaystyle\frac{\Delta y}{\Delta x}$ is the difference quotient (the slope of the secant line).