Editing Difference quotient (section)

{{Short description|Expression in calculus}}
{{broader|Finite difference}}
In single-variable [[calculus]], the '''difference quotient''' is usually the name for the expression

:<math> \frac{f(x+h) - f(x)}{h} </math>

which when taken to the [[Limit of a function|limit]] as ''h'' approaches 0 gives the [[derivative]] of the [[Function (mathematics)|function]] ''f''.<ref name="LaxTerrell2013">{{cite book|author1=Peter D. Lax|author2=Maria Shea Terrell|title=Calculus With Applications|year=2013|publisher=Springer|isbn=978-1-4614-7946-8|page=119}}</ref><ref name="HockettBock2005">{{cite book|author1=Shirley O. Hockett|author2=David Bock|title=Barron's how to Prepare for the AP Calculus|year=2005|publisher=Barron's Educational Series|isbn=978-0-7641-2382-5|page=[https://archive.org/details/isbn_9780764177668/page/44 44]|url-access=registration|url=https://archive.org/details/isbn_9780764177668/page/44}}</ref><ref name="Ryan2010">{{cite book|author=Mark Ryan|title=Calculus Essentials For Dummies|year=2010|publisher=John Wiley & Sons|isbn=978-0-470-64269-6|pages=41–47}}</ref><ref name="NealGustafson2012">{{cite book|author1=Karla Neal|author2=R. Gustafson|author3=Jeff Hughes|title=Precalculus|year=2012|publisher=Cengage Learning|isbn=978-0-495-82662-0|page=133}}</ref> The name of the expression stems from the fact that it is the [[quotient]] of the [[Difference (mathematics)|difference]] of values of the function by the difference of the corresponding values of its argument (the latter is (''x'' + ''h'') - ''x'' = ''h'' in this case).<ref name="Comenetz2002">{{cite book|author=Michael Comenetz|title=Calculus: The Elements|year=2002|publisher=World Scientific|isbn=978-981-02-4904-5|pages=71–76 and 151–161}}</ref><ref name="Pasch2010">{{cite book|author=Moritz Pasch|title=Essays on the Foundations of Mathematics by Moritz Pasch|year=2010|publisher=Springer|isbn=978-90-481-9416-2|page=157}}</ref> The difference quotient is a measure of the [[average]] [[rate of change (mathematics)|rate of change]] of the function over an [[Interval (mathematics)|interval]] (in this case, an interval of length ''h'').<ref name="WilsonAdamson2008">{{cite book|author1=Frank C. Wilson|author2=Scott Adamson|title=Applied Calculus|year=2008|publisher=Cengage Learning|isbn=978-0-618-61104-1|page=177}}</ref><ref name="RubySellers2014"/>{{rp|237}}<ref name="HungerfordShaw2008">{{cite book|author1=Thomas Hungerford|author2=Douglas Shaw|title=Contemporary Precalculus: A Graphing Approach|year=2008|publisher=Cengage Learning|isbn=978-0-495-10833-7|pages=211–212}}</ref> The limit of the difference quotient (i.e., the derivative) is thus the [[instantaneous]] rate of change.<ref name="HungerfordShaw2008"/>

By a slight change in notation (and viewpoint), for an interval [''a'', ''b''], the difference quotient

:<math> \frac{f(b) - f(a)}{b-a}</math>

is called<ref name="Comenetz2002"/> the mean (or average) value of the derivative of ''f'' over the interval [''a'', ''b'']. This name is justified by the [[mean value theorem]], which states that for a [[differentiable function]] ''f'', its derivative ''{{prime|f}}'' reaches its [[Mean of a function|mean value]] at some point in the interval.<ref name="Comenetz2002"/> Geometrically, this difference quotient measures the [[slope]] of the [[secant line]] passing through the points with coordinates (''a'', ''f''(''a'')) and  (''b'', ''f''(''b'')).<ref name="Krantz2014">{{cite book|author=Steven G. Krantz|title=Foundations of Analysis|year=2014|publisher=CRC Press|isbn=978-1-4822-2075-9|page=127}}</ref>

Difference quotients are used as approximations in [[numerical differentiation]],<ref name="RubySellers2014">{{cite book|author1=Tamara Lefcourt Ruby|author2=James Sellers|author3=Lisa Korf |author4=Jeremy Van Horn |author5=Mike Munn|title=Kaplan AP Calculus AB & BC 2015|year=2014|publisher=Kaplan Publishing|isbn=978-1-61865-686-5|page=299}}</ref> but they have also been subject of criticism in this application.<ref name="GriewankWalther2008">{{cite book|author1=Andreas Griewank|author2=Andrea Walther|author2-link=Andrea Walther|title=Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Second Edition|url=https://books.google.com/books?id=qMLUIsgCwvUC&pg=PA2|year=2008|publisher=SIAM|isbn=978-0-89871-659-7|pages=2–}}</ref>

Difference quotients may also find relevance in applications involving [[Temporal discretization |Time discretization]], where the width of the time step is used for the value of h.

The difference quotient is sometimes also called the '''Newton quotient'''<ref name="Krantz2014"/><ref name="Lang1968">{{cite book|author=Serge Lang|title=Analysis 1|url=https://archive.org/details/analysisi0000lang|url-access=registration|year=1968|publisher=Addison-Wesley Publishing Company|page=[https://archive.org/details/analysisi0000lang/page/56 56]|author-link=Serge Lang}}</ref><ref name="Hahn1994">{{cite book|author=Brian D. Hahn|title=Fortran 90 for Scientists and Engineers|year=1994|publisher=Elsevier|isbn=978-0-340-60034-4|page=276}}</ref><ref name="ClaphamNicholson2009">{{cite book|author1=Christopher Clapham|author2=James Nicholson|title=The Concise Oxford Dictionary of Mathematics|url=https://archive.org/details/conciseoxforddic00clap|url-access=limited|year=2009|publisher=Oxford University Press|isbn=978-0-19-157976-9|page=[https://archive.org/details/conciseoxforddic00clap/page/n312 313]}}</ref> (after [[Isaac Newton]]) or '''Fermat's difference quotient''' (after [[Pierre de Fermat]]).<ref>Donald C. Benson, ''A Smoother Pebble: Mathematical Explorations'', Oxford University Press, 2003, p. 176.</ref>