Tangent Line Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Tangent Line.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A line that touches a curve at exactly one point and has the same slope as the curve there.

The tangent line is the unique straight line that best approximates the curve at a specific point — same value, same slope.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The tangent line's slope equals the derivative at that point — it is the best linear approximation to the curve there.

Common stuck point: A tangent can cross the curve elsewhere—it only 'touches' at the point of tangency.

Sense of Study hint: When asked to find a tangent line, first compute f(a) to get the point, then compute f'(a) to get the slope. Finally, plug both into point-slope form: y - f(a) = f'(a)(x - a). Simplify to slope-intercept form if requested.

Worked Examples

Example 1

easy
Find the equation of the tangent line to f(x) = x^2 + 3 at x = 2.

Solution

  1. 1
    Find the point of tangency: f(2) = 4 + 3 = 7. Point: (2, 7).
  2. 2
    Find the slope: f'(x) = 2x, so f'(2) = 4.
  3. 3
    Write the tangent line using point-slope form: y - 7 = 4(x - 2).
  4. 4
    Simplify: y = 4x - 8 + 7 = 4x - 1.

Answer

y = 4x - 1
Two pieces of information define a line: a point and a slope. The point is (a, f(a)) and the slope is f'(a). Plug both into the point-slope formula y - f(a) = f'(a)(x - a).

Example 2

medium
Find the equation of the tangent line to g(x) = \sqrt{x} at x = 9, and use it to approximate \sqrt{9.1}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find the equation of the tangent line to h(x) = 3x^3 - x at x = 1.

Example 2

hard
At what point(s) on f(x) = x^3 - 3x does the tangent line have slope 9?
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Related Concepts

SlopeDerivative

Background Knowledge

These ideas may be useful before you work through the harder examples.

slopederivative