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2.4 Slope and Derivative

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


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