AP®MathsCollege BoardCalculus ABStudy GuidesUnit 2: Differentiation Definition & Fundamental PropertiesDefinition of DifferentiationDerivatives & Tangents

Derivatives & Tangents (College Board AP® Calculus AB): Study Guide

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on

College Board Calculus ABCollege Board Calculus BC

Derivatives and tangents

What is the derivative of a function?

  • The derivative of a function describes the instantaneous rate of change of a function at any given point

    • It is equal to the slope of the curve at that point

  • The derivative of the function f is defined by

    • f to the power of apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of fraction numerator f open parentheses x plus h close parentheses minus f open parentheses x close parentheses over denominator h end fraction

    • This is only valid for values of x where this limit exists

    • Note that the derivative is also a function of x

  • For example, f apostrophe open parentheses 2 close parentheses represents the instantaneous rate of change of the function space f at the point x equals 2

  • There are several ways to denote the derivative of f open parentheses x close parentheses

    • f to the power of apostrophe open parentheses x close parentheses

    • fraction numerator d y over denominator d x end fraction

    • y to the power of apostrophe

  • You may also see "the derivative of ..." written as " ... differentiated"

    • They mean the same thing

  • A tangent line to a curve at a given point is a line that just touches the curve at the point but doesn't cut it at or near that point

    • However, it may cut the curve somewhere else

A graph showing a curve and a tangent line touching the curve at one point with labels explaining that the tangent may intersect and cut the curve elsewhere.

  • The value of the derivative of a function at a point is equal to the slope of the tangent to the graph at that point

    • For example, f apostrophe open parentheses 2 close parentheses represents the slope of the tangent line to the graph y equals f open parentheses x close parentheses at the point x equals 2

Graph of y = f(x) with a curved red line and a blue tangent at x = a, labelled that f′(a) is the slope of the tangent line.
An illustration of the link between the slope of a tangent line and the derivative of a function

When will tangent lines be horizontal or vertical?

  • If the tangent line to the graph of a function f is horizontal, then f to the power of apostrophe open parentheses x close parentheses equals 0

    • Horizontal lines have a slope of zero

  • If the tangent line to the graph of a function f is vertical, then f to the power of apostrophe open parentheses x close parentheses is undefined

    • E.g. f open parentheses x close parentheses equals x to the power of 1 third end exponent with derivative f to the power of apostrophe open parentheses x close parentheses equals fraction numerator 1 over denominator 3 x to the power of 2 over 3 end exponent end fraction

      • f to the power of apostrophe open parentheses 0 close parentheses equals 1 over 0, which is undefined, even though f open parentheses x close parentheses is defined at x equals 0

      • Therefore the graph of f has a vertical tangent at x equals 0

Graph of y = f(x) showing vertical tangent at x = a with f′(a) undefined and horizontal tangent at x = b with f′(b) = 0 on labelled axes
Example of a curve with horizontal and vertical tangent lines

  • But be careful, as there are other reasons a derivative might not be defined at a point

    • E.g. the derivative of g open parentheses x close parentheses equals open vertical bar x close vertical bar is undefined at x equals 0, because the left- and right-hand limits defining the derivative at that point are not equal

    • The graph of g does not have a vertical tangent (or any tangent) at that point

  • Also don't confuse vertical tangents with vertical asymptotes

    • Tangents and curves intersect, but curves only approach asymptotes without ever intersecting with them

    • E.g. h open parentheses x close parentheses equals 1 over x has a vertical asymptote at x equals 0

    • But the function is not defined when x equals 0, so it has no tangent at that point

How do I find the equation of a tangent to a curve using a derivative?

  • To find the equation of a tangent line to the graph of a function f at the point open parentheses a comma b close parentheses using a derivative:

    • Represent the equation of the tangent using the general form for the equation of a straight line with slope m that goes through point open parentheses x subscript 1 comma space y subscript 1 close parentheses

      • y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses

    • Substitute in open parentheses a comma space b close parentheses as the point open parentheses x subscript 1 comma space y subscript 1 close parentheses

      • This is the point the tangent touches on the curve

    • Find the value of the derivative of f open parentheses x close parentheses at the point open parentheses a comma space b close parentheses if it is not given; this is f to the power of apostrophe open parentheses a close parentheses

      • The value of the derivative of f open parentheses x close parentheses at open parentheses a comma space b close parentheses is equal to the slope of the tangent at open parentheses a comma space b close parentheses

    • Substitute in f to the power of apostrophe open parentheses a close parentheses as the value of m in the equation of the tangent

    • The equation of the tangent will be of the form

      • y minus b equals f to the power of apostrophe open parentheses a close parentheses space open parentheses x minus a close parentheses

    • This equation can then be rearranged to another desired form if needed

      • For example, y equals b plus f to the power of apostrophe open parentheses a close parentheses space open parentheses x minus a close parentheses

Worked Example

Let the function f be defined by f open parentheses x close parentheses equals x squared minus 3 x minus 4. It is known that at the point where x equals 4, the instantaneous rate of change of f open parentheses x close parentheses is 5.

Find the equation of the line that is tangent to the graph of f at the point where x equals 4.

Answer:

The tangent is a straight line of the form y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses

The question states that the instantaneous rate of change (the slope) of the curve when x equals 4, is 5

f apostrophe open parentheses 4 close parentheses equals 5

This means the slope of the tangent, m, will also be 5 at this point

The x-coordinate of the point is known, but not the y value

open parentheses 4 comma space y subscript 1 close parentheses

Find the y value by substituting x equals 4 into f open parentheses x close parentheses

y subscript 1 equals f open parentheses 4 close parentheses equals 4 squared minus 3 open parentheses 4 close parentheses minus 4 equals 0

So the point where the tangent touches the curve is open parentheses 4 comma space 0 close parentheses

Substitute the point, and the slope at this point, into the equation of the tangent

y minus 0 equals 5 open parentheses x minus 4 close parentheses

Simplify

y equals 5 open parentheses x minus 4 close parentheses or y equals 5 x minus 20

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

I would just like to say a massive thank you for putting together such a brilliant, easy to use website.I really think using this site helped me secure my top gradesin science and maths. You really did save my exams! Thank you.

Beth
IGCSE Student

This website is soooo useful and I can’t ever thank you enough for organising questions by topic like this. Furthermore, the name of the website could not have been more appropriate as it literally did SAVE MY EXAMS!

Fathima
A Level Student

Incredible! SO worth my money, the revision notes have everything I need to know and are so easy to understand. I actually enjoy revising! It makes me feel a lot more confident for my GCSEs in a few months.

Kate
GCSE Student

Absolutely brilliant, both my girls used it for A levels and GCSE. It's saves on paper copies, also beneficial exam questions ranked from easy to hard. It's removed a lot of stress from the exams.

Sameera
Parent

Just to say that your resources are the best I have seen and I have been teaching chemistry at different levels for about 40 years

Mark
Chemistry Teacher

Excellent

Our rating represented in Trustpilot stars
Read more reviews
Test yourself
Previous:Instantaneous Rate of ChangeNext:Estimating the Derivative at a Point
Author
Reviewer
Jamie Wood

Author: Jamie Wood

Expertise: Maths Content Creator

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths Subject Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.