Introducing Derivatives

    The first main idea of calculus is of course, the limit. A limiting process can be used in the study of curves in general; but the derivative the main limiting process that has lead to the development of calculus.
    Given a function introducing derivatives _gr_1.gif] and a positive introducing derivatives _gr_2.gif], the expression introducing derivatives _gr_3.gif] is called the difference quotient and is the formula for the slope of a secant line to the graph of introducing derivatives _gr_4.gif] through the points introducing derivatives _gr_5.gif] and introducing derivatives _gr_6.gif] The limiting process illustrated in the above example for computing the slope of a tangent line was first developed by the French mathematician Pierre de Fermat. The following definition was realized by Newton and Liebniz.  

Definition (Derivative) The derivative of a function introducing derivatives _gr_7.gif] for any introducing derivatives _gr_8.gif] is given by

introducing derivatives _gr_9.gif]

provided this limit exists. The derivative is also denoted by introducing derivatives _gr_10.gif] and other common notations are introducing derivatives _gr_11.gif], introducing derivatives _gr_12.gif], and introducing derivatives _gr_13.gif] Also the limit is sometimes denoted by

introducing derivatives _gr_14.gif]

The process of finding the derivative is called differentiation. A function introducing derivatives _gr_15.gif] is differentiable at introducing derivatives _gr_16.gif] when the defining limit exists; and we say that introducing derivatives _gr_17.gif] is differentiable on introducing derivatives _gr_18.gif] when introducing derivatives _gr_19.gif] is differentiable at every point in introducing derivatives _gr_20.gif]

Example (Derivative) Find the derivative function using the definition of the derivative. Find the derivative of the function introducing derivatives _gr_21.gif] at introducing derivatives _gr_22.gif]

    Solution. By definition we compute the limit, as follows,
    
introducing derivatives _gr_23.gif]

introducing derivatives _gr_24.gif]

introducing derivatives _gr_25.gif]

introducing derivatives _gr_26.gif]

introducing derivatives _gr_27.gif]

which is the derivative of introducing derivatives _gr_28.gif] for any introducing derivatives _gr_29.gif] At introducing derivatives _gr_30.gif] we have introducing derivatives _gr_31.gif] introducing derivatives _gr_32.gif]

Example (Derivative) Find the derivative function using the definition of the derivative. Find the derivative of the function introducing derivatives _gr_33.gif] at introducing derivatives _gr_34.gif]

    Solution. By definition we compute the limit, as follows,
    
introducing derivatives _gr_35.gif]

introducing derivatives _gr_36.gif]

introducing derivatives _gr_37.gif]

introducing derivatives _gr_38.gif]

introducing derivatives _gr_39.gif]

introducing derivatives _gr_40.gif]

which is the derivative of introducing derivatives _gr_41.gif] for any introducing derivatives _gr_42.gif] At introducing derivatives _gr_43.gif] we have introducing derivatives _gr_44.gif] (Notice that we made use of the formula introducing derivatives _gr_45.gif] introducing derivatives _gr_46.gif]). introducing derivatives _gr_47.gif]

Proposition (Tangent Line) The slope of a tangent line to the function at introducing derivatives _gr_48.gif] is given by   introducing derivatives _gr_49.gif] and the equation of the tangent line is given by

introducing derivatives _gr_50.gif]

Example (Tangent Line) Find the equation of the tangent line at the indicated point. Find the equation of the tangent line to the graph of introducing derivatives _gr_51.gif] at introducing derivatives _gr_52.gif]

    Solution. To find the slope of the tangent line we compute the derivative of the function,
    
introducing derivatives _gr_53.gif]

introducing derivatives _gr_54.gif]

introducing derivatives _gr_55.gif]

introducing derivatives _gr_56.gif]

introducing derivatives _gr_57.gif]

introducing derivatives _gr_58.gif]
    
introducing derivatives _gr_59.gif]

which is the derivative of the function introducing derivatives _gr_60.gif] for any introducing derivatives _gr_61.gif] At introducing derivatives _gr_62.gif] we have

introducing derivatives _gr_63.gif]

Therefore, the equation of the tangent line at introducing derivatives _gr_64.gif] is

introducing derivatives _gr_65.gif]

introducing derivatives _gr_66.gif]

As an illustration the graph of the given function and the tangent line at introducing derivatives _gr_67.gif] is

introducing derivatives _gr_68.gif]
introducing derivatives _gr_69.gif]

Example (Tangent Line) Find the equation of the tangent line at the indicated point. Find the equation of the tangent line to the graph of introducing derivatives _gr_70.gif] at introducing derivatives _gr_71.gif]

    Solution. To find the slope of the tangent line we compute the derivative of the function,
    
introducing derivatives _gr_72.gif]

introducing derivatives _gr_73.gif]

introducing derivatives _gr_74.gif]

introducing derivatives _gr_75.gif]

introducing derivatives _gr_76.gif]

introducing derivatives _gr_77.gif]

which is the derivative of the function introducing derivatives _gr_78.gif] for any introducing derivatives _gr_79.gif] At introducing derivatives _gr_80.gif] we have

introducing derivatives _gr_81.gif]

Therefore, the equation of the tangent line at introducing derivatives _gr_82.gif] is

introducing derivatives _gr_83.gif]

introducing derivatives _gr_84.gif]

As an illustration the graph of the given function and the tangent line at introducing derivatives _gr_85.gif] is


introducing derivatives _gr_86.gif]

introducing derivatives _gr_87.gif]

Cite this as:
Introducing Derivatives
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/introducing-derivatives.html
 
    
Library of Math
Library of Math
Online Math Organized by Subject Into Topics
math search
Library of Math AddThis Feed Button
Online Math
Math Books
Custom math gifts on sale now.
Share your enthusiasm for mathematics!
Be Happy Do Math
Math Gifts
The Library of Math - Online Math Organized by Subject Into Topics.
© 2005 - 2008 www.LibraryOfMath.com All rights reserved.
about us | feedback | privacy policy | terms of use | mision statement | help

Page copy protected against web site content infringement by Copyscape Valid CSS! Valid HTML 4.01 Transitional Subscribe to the Library of Math Feed