Tangent lines

Math

Definition

A tangent line is a straight line that touches a curve at exactly one point and has the same slope as the curve at that point. In calculus, the slope of the tangent line at a point equals the derivative of the function evaluated at that point.

How It Works

  1. Find the derivative f'(x) of the function.
  2. Evaluate f'(a) at the point of tangency x = a to get the slope.
  3. Find the y-coordinate by evaluating f(a).
  4. Write the equation using point-slope form: y − f(a) = f'(a)(x − a).

Examples

  • The tangent line to y = x² at x = 3 has slope 6, giving y = 6x − 9
  • Finding the tangent line to determine instantaneous velocity on a position graph
  • Using a tangent line to approximate √4.1 via linearization
Key Fact

Tangent line at x = a: y − f(a) = f'(a)(x − a)

Related Terms

Derivative definition Linearization Point-slope form Implicit differentiation

Study This Concept

Practice tangent lines with free review games in these units:

AP Calculus AB Unit 2: Differentiation: Definition Geometry Unit 9: Circles