# 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?

• I don't know about (2), but I'd agree with you on (1) and (3). Commented Aug 11, 2013 at 1:44
• Somewhat of a duplicate of What's the difference between $\frac{\delta}{dt}$ and $\frac{d}{dt}$? Commented Aug 11, 2013 at 1:45
• @Haizum: Are you sure you mean $\dfrac{\delta y}{\delta x}$ and not $\dfrac{\partial y}{\partial x}$? Commented Aug 11, 2013 at 1:46
• I'd say that you are right on (2) as well, although it is primarily used to denote directional derivative for functions on infinite-dimensional spaces (in calculus of variations). Commented Aug 11, 2013 at 1:48
• Here is a (somewhat) useful link with review of various notation: en.wikipedia.org/wiki/Notation_for_differentiation Commented Aug 11, 2013 at 1:50

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).