Find study content
Learning Materials

Discover learning materials by subject, university or textbook.

Explanations

Textbooks

Exams
All Subjects

Anthropology

Archaeology

Architecture

Art and Design

Bengali

Biology

Business Studies

Greek

Chemistry

Chinese

Combined Science

Computer Science

Economics

Engineering

English

English Literature

Environmental Science

French

Geography

German

History

Hospitality and Tourism

Human Geography

Italian

Japanese

Law

Macroeconomics

Marketing

Math

Media Studies

Medicine

Microeconomics

Music

Nursing

Nutrition and Food Science

Physics

Polish

Politics

Psychology

Religious Studies

Sociology

Spanish

Sports Sciences

Translation

All Subjects

Biology

Business Studies

Chemistry

Combined Science

Computer Science

Economics

English

Environmental Science

Geography

History

Math

Physics

Psychology

Sociology

EXAM TYPES

GCSE

IGCSE

AS

A Level

International A Level

University Admissions Tests

GCSE SUBJECTS

GCSE 1

GCSE 2

GCSE 3

GCSE 4

GCSE 5

SOME-TEXT

IGCSE SUBJECTS

IGCSE 1

AS SUBJECTS

AS 1

A Level SUBJECTS

A Level 1

International A Level SUBJECTS

International A Level 1

University Admissions Tests SUBJECTS

University Admissions Tests 1
Features
Features

Discover all of these amazing features with a free account.

Flashcards

Vaia AI

Notes

Study Plans

Study Sets
What’s new?

Flashcards
Study your flashcards with three learning modes.

Study Sets
All of your learning materials stored in one place.

Notes
Create and edit notes or documents.

Study Plans
Organise your studies and prepare for exams.
Resources
Discover

All the hacks around your studies and career - in one place.

Find a job

Find a degree

Student Deals

Magazine

Mobile App
Featured

Magazine
Trusted advice for anyone who wants to ace their studies & career.

Job Board
The largest student job board with the most exciting opportunities.

Vaia Deals
Verified student deals from top brands.

Our App
Discover our mobile app to take your studies anywhere.

Learning Materials

Features

Discover

Tangent Lines

In Latin, the word tangent means "to touch." So then, a tangent line is a line that touches. Consider a bicycle moving along the flat pavement. The road is essentially tangent to the bicycle wheel as it touches the wheel at a point. In this article, we will further discuss the meaning of a tangent line, a tangent line's formula, and what the slope of a tangent line means.

Get started

+ Add tag
Immunology
Cell Biology
Mo

The slope of the tangent line at a point P is...

Show Answer

+ Add tag
Immunology
Cell Biology
Mo

The tangent line touches the curve at ___________ point.

Show Answer

+ Add tag
Immunology
Cell Biology
Mo

The slope of the tangent line at a point P is...

Show Answer

+ Add tag
Immunology
Cell Biology
Mo

The tangent line touches the curve at ___________ point.

Show Answer

Fact Checked Content
Last Updated: 25.01.2023
6 min reading time

Content creation process designed by
Content cross-checked by
Content quality checked by

Definition and formula of a Tangent Line

A tangent line is a line that "just touches" the point $P$ . It can also be defined as a line joining 2 infinitely close points on a curve.

The tangent line to a curve $f (x)$ at a point $P$ , with coordinates $(a, f (a))$ , is the line through $P$ with slope

$m = \lim_{x \to a} \frac{f (x) - f (a)}{x - a}$

if the limit exists.

Equation of a Tangent Line

Once the slope $m$ is found, the equation of a tangent line is the same as any other line in point-slope form through a point $(a, f (a))$ :

$(y - f (a)) = m (x - a)$

Tangent Lines on a Graph

In the graph below, we say that $y$ is a tangent line to the curve $f$ at point $P$ . Or, we can say that $y$ is tangent to the curve $f$ at point $P$ .

Tangent Line geometric interpretation Vaia The tangent line in green merely touches the curve f at point P - Vaia Original

Notice how the tangent line "just touches" the curve at point P.

The slope of a tangent line

Geometry

The slope of the tangent line at a point on a curve is equal to the slope of the curve at that point. The assumption behind tangent lines is when looking at the graph of a curve, if you zoom in close enough to a segment of the curve, the curve will look indistinguishable from the tangent line.

For example, let's zoom in on the graph above.

Tangent Lines slope of tangent line equal to curve Vaia Zooming in at the point where the tangent line touches the curve - Vaia Original

Zooming in a bit more...

Tangent Lines slope of tangent line equal to curve Vaia

Here we can see that the tangent line and the point of the curve where the tangent line touches are indistinguishable - Vaia Original

Notice how the tangent line comes from connecting two infinitely close points on a curve.

Additional tangent line slope equation

There is another version of the equation for the slope of the tangent line that can be easier to work with. This equation says that the slope of the tangent line $m$ is

$m = \lim_{h \to 0} \frac{f (a + h) - f (a)}{h}$

This equation sets $h = x - a$ and $x = h + a$ . As $x$ approaches $a$ , $h$ approaches 0. Thus, the supplementary equation of the tangent line is formed.

This equation for the slope of the tangent line should look familiar to you.... It is the equation for the derivative of a function at a point (a, f(a)). So, we can say that the slope of the tangent line on a curve at point P is equal to the derivative of the curve at point P !

Examples of finding the Tangent Line Equation

Example 1

Find the equation for the tangent line to $f (x) = x^{2}$ at the point (2, 4).

As we are given $f (x)$ and a point, all we need to form the equation of the tangent line is the slope. To find the slope, we will use the supplementary equation of the tangent line.

The slope of the tangent at (2, 4) is

$\begin{array}{rcl} m & = & \lim_{h \to 0} \frac{f (2 + h) - f (2)}{h} \\ = & \lim_{h \to 0} \frac{{(2 + h)}^{2} - 4}{h} \\ = & \lim_{h \to 0} \frac{h^{2} + 4 h + 4 - 4}{h} \\ = & \lim_{h \to 0} \frac{h^{2} + 4 h}{h} \\ = & \lim_{h \to 0} \frac{h (h + 4)}{h} \\ = & \lim_{h \to 0} h + 4 \\ = & 4 \end{array}$

So, the equation of the line tangent to f(x) at (2, 4) is $y = 4 (x - 2) + 4$ .

Recall how the slope of the tangent line is the same of the derivative. So, we could also simply take the derivative of $f (x)$ and plug in $x = 2$ to find the slope of the tangent line at the point $(2, 4)$ .

$f' (x) = 2 x f' (2) = 4$

Tangent lines graph of tangent line and function example Vaia The graph of f(x) and the line tangent to f(x) at (2, 4) - Vaia Original

Example 2

Find the equation of the tangent line to the curve $f (x) = x - x^{3}$ at the point (1, 0).

Again, as we are given $f (x)$ and a point, all we need to form the equation of the tangent line is the slope. To find the slope, we will use the supplementary equation of the tangent line.

With , the slope of the tangent at (1, 0) is:

$\begin{array}{rcl} m & = & \lim_{h \to 0} \frac{f (1 + h) - f (1)}{h} \\ = & \lim_{h \to 0} \frac{[(1 + h) - {(1 + h)}^{3}] - 0}{h} \\ = & \lim_{h \to 0} \frac{(1 + h) - (h^{3} + 3 h^{2} + 3 h + 1)}{h} \\ = & \lim_{h \to 0} \frac{h^{3} - 3 h^{2} - 2 h}{h} \\ = & \lim_{h \to 0} h^{2} - 3 h - 2 \\ = & - 2 \end{array}$

So, the equation of the line tangent to f(x) at (1, 0) is $y = - 2 (x - 1)$ .

Again, recall how the slope of the tangent line is the same of the derivative. So, we could also simply take the derivative of $f (x)$ and plug in 1 to find the slope of the tangent line at the point $(1, 0)$ .

$f' (x) = 1 - 3 x^{2} f' (1) = 1 - 3 (1) = - 2$

Tangent Line graph of the line tangent to the curve example Vaia The graph of f(x) and the line tangent to f(x) at (1, 0) - Vaia Original

Tangent Lines in a Circle

A line is said to be tangent to a circle if it touches the circle at exactly one point. If a line is tangent to a circle at a point $P$ , then the tangent line is perpendicular to the radius drawn to point $P$ .

Tangent Lines tangent line circle perpendicular radius Vaia A line that is tangent to a circle at point P is perpendicular to the radius drawn to point P - Vaia Originals

For more information on tangent lines in a circle, check out our article on Tangent of a Circle!

Tangent Lines - Key takeaways

A tangent line is a line that touches a curve at a fixed point $P$
- The equation of a tangent line in point-slope form is $(y - f (a)) = m (x - a)$
The slope of the tangent line at a point on a curve is equal to the slope of the curve at that point
- If you zoom in close enough to a segment of a curve, the tangent line at the segment and the curve will look indistinguishable
In a circle, the tangent line drawn at any point is perpendicular to the radius at that point.

Flashcards in Tangent Lines

Start learning

The slope of the tangent line at a point P is...

equivalent to the slope of the curve at P

The tangent line touches the curve at ___________ point.

one

Already have an account? Log in

Frequently Asked Questions about Tangent Lines

How do you find tangent line?

To find a tangent line, use the equation for the slope of a tangent line at a point and plug in the slope and the point into point-slope form.

What is tangent line of a curve?

A tangent line of a curve is a line that touches the curve at a fixed point P.

What defines a tangent line?

A tangent line is a line that touches the curve at a fixed point P. The slope of the point P at the curve is equal to the slope of the tangent line at point P.

How to find the equation of line tangent?

To find the equation of a tangent line, use the equation for the slope of a tangent line at a point and plug in the slope and the point into point-slope form.

How to find slope of a tangent line?

The slope of a tangent line can be found using the limit definition of the slope of a tangent line.

Save Article

How we ensure our content is accurate and trustworthy?

At StudySmarter, we have created a learning platform that serves millions of students. Meet the people who work hard to deliver fact based content as well as making sure it is verified.

Content Creation Process:

Lily Hulatt is a Digital Content Specialist with over three years of experience in content strategy and curriculum design. She gained her PhD in English Literature from Durham University in 2022, taught in Durham University’s English Studies Department, and has contributed to a number of publications. Lily specialises in English Literature, English Language, History, and Philosophy.

Get to know Lily

Content Quality Monitored by:

Gabriel Freitas is an AI Engineer with a solid experience in software development, machine learning algorithms, and generative AI, including large language models’ (LLMs) applications. Graduated in Electrical Engineering at the University of São Paulo, he is currently pursuing an MSc in Computer Engineering at the University of Campinas, specializing in machine learning topics. Gabriel has a strong background in software engineering and has worked on projects involving computer vision, embedded AI, and LLM applications.

Get to know Gabriel

Discover learning materials with the free Vaia app

About Vaia

Vaia is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

Learn more

Vaia Editorial Team

Team Math Teachers