What does the derivative of a function at a point describe?

[dustball]

To give an alternate non-calculus viewpoint we can use a very old fashioned statement. The derivative is the slope of the line/plane that intercepts a curve/surface at exactly one-point. If the line isn't unique then the "derivative" isn't unique. It is presumed that the line is as short as necessary to avoid bumps.

Sorry, this definition does not work, consider the derivative of $x^3$ at $x=0$ or $x^2sin(x^{-2})$ (taken to be 0 at 0) at $x=0$.

 

[dustball]

But endeed, the full power of classical analysis and limit theory is not needed to understand differentiation. It can be done on a more elementary level. Unfortunately not too many people are aware of it.