CALCULUSTHOMAS'CALCULUSARUYTRANSCENChapter 3JAMESSTEWARTDerivativesAAStCiHA3.11.Definitionof DerivativeTheDerivative as aFunction嘉Slide 3-4Derivative of functionatonepointDerivative offunctionDEFINITIONThe derivative ofthe function fix) with respect to the variablexThe derivative off(x) at point x = is the slope ofcurveis thefunction f"whose value at x is(x+h)- f(r)(x +h)f(x)y=f(x) at xo, whose value is lim '(x)h1provided the limit exissFis differentiablef(x)(has aderivative)at xThe domain of f'is the set of points in the domain of fforwhich the limit exists. Its domain may be the same as thedomainofforitmaybesmallerIf f'exists at every point ofthedomain off,we call fisdifferentiableSlide3-5slide 3-8
2016/11/15 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 3 Derivatives Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.1 The Derivative as a Function Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 3 - 4 1. Definition of Derivative Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 5 Derivative of function at one point The derivative of f(x) at point is the slope of curve y=f(x) at , whose value is 0 x x 0 x 0 0 0 ( ) ( ) lim h f x h f x h 0 f x'( ) f is differentiable (has a derivative) at 0 x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 6 Derivative of function The domain of f ' is the set of points in the domain of f for which the limit exists. Its domain may be the same as the domain of f or it may be smaller. If f ' exists at every point of the domain of f , we call f is differentiable
EXAMPLECalculatingf'(x)fromtheDefinitionofDerivative(a) Find the derivative of f(x) = V for x > 0.(b) Find the tangent line to the curve y =Vx at x = 4Step 1 Write expressions forf(x) and f(x+h)ExercisesStep 2 Expand and simplify the difference quotientFinding Derivative Functions and ValuesUsing the definition, calculate the derivatives ofthe functions in Exer(x+h)-J(x)cises 1-6. Then find the values of the derivatives as specified.h1. f(x) = 4 -x; r(3), r(0). f(1)Step 3 Using the simplified quotient, find f(x) by2. F(x) = (x - 1) + 1; F(1), F(0), F(2)evaluating the limit0)=m a+h-)3. g(0) =: g(-1.g(2).g(V)hR(1),R(1), R(V2)4. 8(2) 2;5. p(0) = V30 : p(1),p(3),p'(2/3)Slide3-7Duu3-8>the processofcalculatingaderivativeiscalleddifferentiatioy(x,c)(x +h,c).002.Differentiation of a Constants,Powers,Multiples,andSumsx0x+hFIGURE3.6Therule (d/dx)(c)=0isanother way to say that the values ofconstant functions never change and thatthe slope of a horizontal line is zero atevery point.Slide 3-10Rule2PowerRuleforPositiveIntegersRULE1Derivative of a Constant FunctionIf j has the constant value f(r) c, thendx"= -If n is a positive integer, then -(0 0:dxSlide3-11Slide 3-12
2016/11/15 2 Slide 3 - 7 Calculating f'(x) from the Definition of Derivative Step 1 Write expressions for f(x) and f(x+h) Step 2 Expand and simplify the difference quotient f x h f x ( ) ( ) h 0 ( ) ( ) '( ) limh f x h f x f x h Step 3 Using the simplified quotient, find f'(x) by evaluating the limit Slide 3 - 8 Exercises Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2. Differentiation of a Constants, Powers, Multiples, and Sums the process of calculating a derivative is called differentiation Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 12 Rule 2 Power Rule for Positive Integers If n is a positive integer, then d n n 1 x nx dx
RULE4Derlvative Sum RuleIf u and are differentiable functions ofx, then their sum w + is differentiableat every point where w.and u are both differentiable.At such points,RULE3Constant Muttiple RuleIfr is a differentiablefunction ofx,and c is aconstant, then+）一+（）尝CoExtensionform ofDerivativeSumRuleausefulspacialThe Sum Rule also extends to sum ofmore than two functions,ddu(-u)=-as long as there are only finitely many functions in the sum.dxdxIf, 12,*are differentiableatx, then so is ,+,+andd+u+,+..dxSlide3-13Slide3-14EXAMPLE4Does the curve y xd 2x? + 2 have any borizoetat tangents? If soExerciseswhere?y=x*_ 2x+2Siopes.and Tangent LinesIn Exercises 13-16, differentiate the functions and find the slope ofthe tangent line at the given value of the independent variable.-+量x--314-20.216+5-81--11X=-2In Excrcises178,diferentiate thefunctions,Thnfind anequaionofthctangetlineattheindicated pointothegraphofthefumction(1, 1)(1..1)f17. y= J(o) =6=(6,4)Vx-2*618 w= g(2) = 1 + V4 , (w) = (3,2)FIGURE3.8 The curvey =42r? + 2 and its horizontaltangents (Example 4).Slide 3- 15Slide 3-18Right-handSlope =derivativeatam ib+ h f(b)fh-0SlopeLeft-handf(a + h) - J(a)limderivativeatb/3.DeferentiableonanInterval;y =f(x)One-SidedDerivativesa+hb+hah<0h>0FIGURE3.3Derivatives at endpoints areone-sided limits.Slide3-17slide 3-18
2016/11/15 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 13 a useful special case ( ) d du u dx dx Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 14 Extension form of Derivative Sum Rule The Sum Rule also extends to sum of more than two functions, as long as there are only finitely many functions in the sum. If are differentiable at x, then so is , and 1 2 , , , n u u u 1 2 n u u u 1 2 1 2 ( ) n n d du du du u u u dx dx dx dx Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 15 Slide 3 - 16 Exercises Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 3 - 17 3. Deferentiable on an Interval; One-Sided Derivatives Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 18 Left-hand derivative at b Right-hand derivative at a
Theusual relationbetweenone-sidedandAfunctiony=f(x)isdifferentiableonanopenintervaltwo-sided limits holds forthesederivatives.(finite or infinite)if it has a derivativeat each pointofthe interval. It is differentiable on a closed interval [a,Afunction has a derivativeata pointif andb) if it is differentiable on the interval (a,b)and if itonly if it has left-hand and right-handhasderivativesatendpoints aandbderivatives there,and these one-sidedderivatives are equal.Slide3-19Slide 3-20One-Ssded Derhiative:*In generalif thegraphofExercisesComoste thenghe-hand and keft-hond dcrivatives as limts 1o show thathe functioos inExercises3740arenotdifferettableatthepointa function has a 'cormer,37.38.then there is no tangent atthis potnt and fis notdifferentiable there, Thus,differentabiliy is a"smoothness"condition.y'not defined at x = O:right-hand derivativeA# left-hand derivativeFIGURE 3.4 The function y =[xisnot differentiable at the origin wherethe graph has a "corner"Slide3-21Slide 3- 22What can.we learn from the.graph of y. I'(x)?.At a glance we can sewheretherateofchangef ispositive,negative,or zeo1,2.therough size ofthegrowthrateatanyxand its sizeinrelation to the size of fcx):3,wheretherateofchange itselfisincreasing ordecreasing4.DifferentiableFunctionsAreContinuousSlide 3-23Slide 3-24
2016/11/15 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 19 A function y=f(x) is differentiable on an open interval (finite or infinite) if it has a derivative at each point of the interval. It is differentiable on a closed interval [a, b] if it is differentiable on the interval (a, b) and if it has derivatives at endpoints a and b. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 20 The usual relation between one-sided and two-sided limits holds for these derivatives. A function has a derivative at a point if and only if it has left-hand and right-hand derivatives there, and these one-sided derivatives are equal. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 21 In general, if the graph of a function has a "corner", then there is no tangent at this point and f is not differentiable there. Thus, differentiability is a "smoothness" condition. Slide 3 - 22 Exercises Slide 3 - 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 3 - 24 4. Differentiable Functions Are Continuous
Differentiabflity and Continuity on an IntervaEach figure im Exercises 23-26 shows the ghiph of a function over aclosed imerval D.Atwtat domuin points does the fanetion appeartobeExercisesa. differentisble?THEOREM1Differentiability Implies Continuitybcortimousbutnotdifferentiable?If f has a derivativeat x c, then J iscontinuous atxcDos mor differentiable?Eneither conGIW学Iffhasaderivativefromoneside (right orleft)atx=c23thenfiscontinuousfromthatsideatx=cIf the function has a discontinuity at a point (for.instance, a jump discontinuity), then it cannot bedifferentiablethere.slide3-25Slide 3-28THEOREM2Darboux's Theorem5.Intermediate ValuepropertyIf a and b are any two points in an interval on which f is differentiable, then ftakes on every value between f"(a) and f"(b).of Derivatives3Theorem2 says that a function cannot be a derivativeon an interval unless it has the Intermediate ValuePropertythere.Slide 3-27Slide 3- 28y=U(x)6.Second-andHigher-OrderDerivatives0FIGURE3.5The unit stepWe cannot find a function f(x)function does not have thedefined on the intervalIntermediate Value Property and(-m,+) such that U(x)=f()cannot be the derivative of afunction on the real lineSlide 3- 29slide 3-30
2016/11/15 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 25 If f has a derivative from one side (right or left) at x=c then f is continuous from that side at x=c. If the function has a discontinuity at a point (for instance, a jump discontinuity), then it cannot be differentiable there. Slide 3 - 26 Exercises 23-26 23. 24. 25. 26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 3 - 27 5. Intermediate Value property of Derivatives Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 28 Theorem 2 says that a function cannot be a derivative on an interval unless it has the Intermediate Value Property there. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 29 We cannot find a function f(x) defined on the interval (-∞,+∞) such that U(x)=f'(x) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 3 - 30 6. Second- and Higher-Order Derivatives