What a Derivative Really Measures

Lesson notes

Key points

• A derivative measures the instantaneous rate of change of a function at a single point.
• We start with the slope of a secant line through two nearby points on the curve.
• Letting the gap h shrink toward zero turns the secant slope into the tangent slope, the derivative.
• For f of x equals x squared, this limit gives two x, so the slope at x equals three is six.

Definitions

Derivative

The instantaneous rate of change of f(x) at a point; the slope of the tangent line there.

Takeaway: A derivative is the limit of secant slopes: the tangent slope.

DerivativeSecant slopeLimit definitionWorked example