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: A tangent line touches a curve at a point with the same value and the same slope, giving the best linear approximation there.

Common stuck point: The procedure for tangent line is the easy part; the trap is using the slope of a secant instead of $f'(a)$ . Asking "Do I need the line that matches the curve's value and slope ( $f'(a)$ ) at a single point of contact?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do I need the line that matches the curve's value and slope ( $f'(a)$ ) at a single point of contact?

Worked Examples

Example 1

easy

Find the equation of the tangent line to

f(x) = x^2 + 3

x = 2

Answer

y = 4x - 1

First step

Find the point of tangency:

f(2) = 4 + 3 = 7

. Point:

(2, 7)

Full solution

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

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}

x = 9

, and use it to approximate

\sqrt{9.1}

Example 3

medium

Find the equation of the tangent to

f(x) = \frac{1}{x}

x = 2

Example 4

medium

At which point on

f(x) = x^2

is the tangent line parallel to

y = 6x - 5

Example 5

medium

Find the tangent line to the curve

f(x) = x^2 - 2x + 3

at the point where the tangent is horizontal.

Example 6

medium

Find the tangent line to

f(x) = x^2

that passes through the external point

(2, -3)

Example 7

hard

Find the equation of the tangent to the implicit curve

x^2 + y^2 = 25

at the point

(3, 4)

Example 8

hard

For

f(x) = x^3 - 6x^2 + 9x

, find values of

a

for which the tangent line at

(a, f(a))

passes through the origin.

Example 9

challenge

Show that the tangent line to

y = x^n

(1, 1)

has

x

-intercept

1 - 1/n

for any

n \ne 0

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

x = 1

Example 2

hard

At what point(s) on

f(x) = x^3 - 3x

f(x) = \frac{1}{x}

x = 2

Example 13

medium

At what point does

f(x) = x^2

have a tangent line parallel to

y = 6x

Example 14

medium

Find the tangent to

f(x) = \cos x

x = \frac{\pi}{2}

Example 15

medium

Use the tangent line of

f(x) = \sqrt{x}

x = 4

to approximate

\sqrt{4.1}

Example 16

medium

Find all points where

f(x) = x^3 - 3x

has a horizontal tangent.

Example 17

medium

Find the tangent to

f(x) = \ln x

x = 1

Example 18

challenge

Find the equations of the tangent lines to

f(x) = x^2

that pass through the external point

(0, -1)

Example 19

challenge

Find where the tangent to

f(x) = x^3

x = a

also crosses the curve again.

Example 20

challenge

The tangent to

f(x) = \frac{1}{x}

f(x) = \sqrt{x}

x = 25

to approximate

\sqrt{26}

Example 31

medium

Find all points where

f(x) = x^3 - 6x^2 + 9x

has a horizontal tangent.

Example 32

medium

Find the equation of the normal line to

f(x) = x^2

x = 1

Example 33

medium

Find the angle the tangent to

f(x) = x^2

x = 1

makes with the positive

x

-axis.

Example 34

hard

Find both points on

f(x) = x^3 - 12x

where the tangent line has slope

0

Example 35

hard

At what point on

f(x) = e^x

is the tangent line parallel to

y = e \cdot x + 7

Example 36

hard

Find the tangent line to

f(x) = x \ln x

x = 1

Example 37

hard

Find the tangent line to the curve

y^2 = x^3 - 3x + 3

at the point

(1, 1)

Example 38

hard

Find the tangent line to the curve

x^2 + xy + y^2 = 7

(1, 2)

Example 39

challenge

A line through

(0, -2)

is tangent to

f(x) = x^2 + 1

. Find both possible slopes.

← Back to Tangent Line Formula Explained Practice Mode

Related Concepts

Slope Derivative

Background Knowledge

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

slopederivative