A function f is differentiable at x₀ if it looks like a straight line (called its tangent line sufficiently near x₀.Its derivative at x₀ is the slope of that line.It is denoted by f '(x₀) or and its formal definition is

Notation

Here  is the slope of the secant line which approaches the slope of the tangent line as Dx approaches 0.

How small must Dx be so that  is f '? Answer

The derivative is f 's "instantaneous" rate of change with respect to variable x.

If x is time and f represents position, then f ' is velocity.

Ball example