MTH 251 – Differential Calculus Chapter 3 – DifferentiationReview – The Derivative at a PointThe Derivative as a FunctionDerivative NotationDerivative at x = a NotationExamples …Slide 7Sketching the Graph of f ’(x) using the Graph of f(x)Slide 9Left & Right Derivatives at a PointSlide 11Where does a derivative NOT exist?Slide 13Slide 14Slide 15Differentiability & ContinuitySlide 17MTH 251 – Differential CalculusChapter 3 – DifferentiationSection 3.2The Derivative as a FunctionCopyright © 2010 by Ron Wallace, all rights reserved.Review – The Derivative at a Point•The derivative was defined as the limit of the difference quotient. That is …•If x0 + h = z, then an alternate definition would be …•Note that the result of this limit is a number. That is, the derivative at a specific value of x.( )0 000( ) ( )' limhf x h f xf xh�+ -=Remember: x0 refers to a specific value of x.( )0000( ) ( )' limz xf z f xf xz x�-=-The Derivative as a Function•If we do not specify a specific value of x (i.e. use x instead of x0) we get a function called the derivative of f(x).•That is, the derivative of f(x) is the function …0( ) ( )'( ) limhf x h f xf xh�+ -=( ) ( )'( ) limz xf z f xf xz x�-=-ORf(x+h) xx+hhf(x)f(x+h) – f(x)Derivative Notation•All of the following can be used to designate the function that is the derivative of y = f(x)'f'( )f x'y[ ]( )df xdx[ ]( )xD f xdydxdfdxReminder: The results of these will be a function.Derivative at x = a Notation•All of the following can be used to designate the derivative of y = f(x) at x = a'( )f a[ ]( )x adf xdx=x adydx=[ ]( )xaD f xadfdxReminder: The results of these will be a number.Examples …•Determine the following derivatives …1ddx x������dxdx� �� �ndxdx� �� � IMPORTANT!Memorize these 3 results.Examples …•Determine the following derivatives …71xddx x=������9dxdx� �� �5dxdx� �� �Sketching the Graph of f ’(x) using the Graph of f(x)•Where is the derivative (i.e. slope) zero?•Where is the derivative (i.e. slope) positive?Large or small positive?•Where is the derivative (i.e. slope) negative?Large or small negative?•Where is the derivative (i.e. slope) constant?Function is a line segment.Derivative is a horizontal line segment.Sketching the Graph of f ’(x) using the Graph of f(x)•Example – Sketch the graph of the derivative of the following function.Left & Right Derivatives at a Point•If in the definition of the derivative at a point, you use just the left or right hand limit, the derivative at a point can be considered from just one side or the other.•Right-Hand Derivative at x0•Left-Hand Derivative at x0•If these are equal, then …( )( ) ( )0 000' limhf x h f xf xh++�+ -=( )( ) ( )0 000' limhf x h f xf xh--�+ -=( ) ( )( )00 0' ' 'f x f x f x+ -= =Left & Right Derivatives at a Point•Example: ( )( )( )24' 2 ?' 2 ?f x xff+-= -==Where does a derivative NOT exist?•Cornerleft & right derivatives are different22y x= -Where does a derivative NOT exist?•Corner•Cuspleft & right derivatives are approaching  & –21524( 1)y x= + -Where does a derivative NOT exist?•Corner•Cusp•Vertical TangentThe derivative limit is  or –31 1y x= + -Where does a derivative NOT exist?•Corner•Cusp•Vertical Tangent•Discontinuitysee the next theorem3 265( 3)x x xyx- -=-Differentiability & Continuity•If f ’(c) exists, then f(x) is continuous at x = c.Proof …Let x c h= +Therefore: lim ( ) ( )x cf x f c�=0lim ( )hf c h�+0lim ( ) [ ( ) ( )]hf c f c h f c�= + + -0[ ( ) ( )]lim ( )hf c h f cf c hh�+ -= + �0 0 0[ ( ) ( )]lim ( ) lim limh h hf c h f cf c hh� � �+ -= + �( ) '( ) 0 ( )f c f c f c= + �= as 0x c h�ޮޮDifferentiability & Continuity•If f ’(c) exists, then f(x) is continuous at x = c.Or … the contrapositive implies …•If f(x) is NOT continuous at x = c, then f ’(c) does not exist.•NOTESIf the derivative does not exist, that does not mean the function is not continuous.If the function is continuous, that does not mean that the derivative exists.Example … the Absolute Value