Derivative definition

Math

Definition

The derivative of a function at a point gives the instantaneous rate of change, or the slope of the tangent line at that point. It is formally defined as the limit of the difference quotient as the interval approaches zero.

Examples

  • The derivative of f(x) = x² is f'(x) = 2x, so at x = 3 the slope is 6
  • Velocity is the derivative of position with respect to time
  • The derivative of a constant function is always 0
Key Fact

f'(x) = lim h→0 [f(x+h) - f(x)] / h

Related Terms

Limit definition Tangent lines Chain rule Implicit differentiation

Study This Concept

Practice derivative definition with free review games in these units:

AP Calculus AB Unit 2: Differentiation: Definition