EXAMPLE10The first four derivatives of y = 3r? + 2 areThederivativey'=dyldxisthefirst(first-orderFirst derivative:y3r26tderivativeofywithrespecttox.Second derivativey'-6r-6Third derivutive:y=6Ifyisa differentiablefunction ofx,its derivativey(40iscalledthesecond(secondFourth derivative:order)derivative ofy with respect to x.The function has derivatives of all orders, the fifth and later derivatives all being zero.d'ythethird (third-order) derivatived*Exercises"=ofywith.respecttoxFind the first and second derivatives of the functions,=2the nth (nth-order) derivativetofywith respectto x2 - (+)3 -2)We can interpret the second derivative as the rate of changeof the slope oftangent to the curve y-f(x)at each point.Slide3-31Slide 3-32a corner, where the one-sided2.cusp,wheretheslopefPI.derivatives differ.approaches o from one side and 0ofrom the other.LawrtioatMangent,1,adiscuuinty(twoexamplesshown)where the slope of Pgapproaches co from bothsides or approachesocfrom both sides (here, 00).Slide3-33Slide 3-34Derivative of the Natural Exponential Functionta.es#=FIGURE 3.10 The line through the originis tangent to the graph ofy mewhena = 1 (Example 5)Slide 3-35Slide 3-38
2016/11/15 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 31 The derivative y'=dy/dx is the first (first-order) derivative of y with respect to x. If y' is a differentiable function of x, its derivative is called the second (secondorder) derivative of y with respect to x. 2 2 ' '' ( ) dy d dy d y y dx dx dx dx 3 3 '' ''' dy d y y dx dx the third (third-order) derivative of y with respect to x. ( ) ( 1) n n d y y dx the nth (nth-order) derivative of y with respect to x. We can interpret the second derivative as the rate of change of the slope of tangent to the curve y=f(x) at each point. Slide 3 - 32 Exercises 2. 1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 33 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 34 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 35 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 36
3.21.Instantaneous Rates of ChangeThe Derivative as aRateof ChangeAslide 3-38DEFINITIONThe instantanrate ofchangeof fwithrespect toxat xo isthe derivative(o+h)-F)f'(xo)2.Motion Along a Lineprovided the limit existsDisplacement,Velocity,SpeedAcceleration,and Jerkthe averagerate of change infovertheirtenvalfromxtoXhSlide 3- 39Slide 3- 40Position at time t.and attimei+ArAsAsmf(t)s+A=f+)DEFINITIONVelocity (instantaneous velocity) is the derivative of positiowith respect to time. if a body's position at time.t is s f(t), then the bodyFIGURE 3.11 The positions ofa bodyvelocity at time fismovingalongacoordinatelineattimet)== lim.and shortly later at time t + r.Ar1The displacement of the object over the timeinterval fromt to t+△t is △s=f(t+△t)-f(t)2Theaverage velocityof theobjectoverthattimeintervalisglacmm+-travelfimeSlide 3-41Slide 3- 42
2016/11/15 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.2 The Derivative as a Rate of Change Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 3 - 38 1. Instantaneous Rates of Change Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 39 the average rate of change in f over the interval from to +h 0 x 0 x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 3 - 40 2. Motion Along a Line: Displacement, Velocity, Speed, Acceleration, and Jerk Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 41 The displacement of the object over the time interval from t to t+Δt is Δs=f(t+Δt)-f(t) The average velocity of the object over that time interval is ( ) ( ) av displacement s f t t f t v travel time t t Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 42
fu8DEFINITIONSpeed is the absolute value of velocitytisoSpeed - [o(0F positive slope sonegativemoving forvardmoving backwaFIGURE3.12For motion s = f(c) along a straight line, u = ds/dt ispositive when s increases and negative when s decreases.RSpeedmeasures the rate of progressWhen the_object is moving forwardregardless of direction.Velocitytells-how(s increasing), the velocity is positive.fastan object isnovingandtheWhen the object is moving backwarddirectionofmotior(s decreasing), the velocity is negatfive.Slide3-43Slide3-44A sudden change in acceleration is called a "jerk.WAERENA020e=ynWhena ride in a car or a bus is jerk, it is not that theacceleration involved are necessarily large but that theaomtchanges inaccelerationare abrupt.still0-0DEFINITIONSAceeleration is the derivative of velocity with respect to timc+fIfyositiontim)tboyacceeratntimThe rate at which a body'sfevelocity changes is thebody's acceleration.Jerk is the derivative of acceleration with respect to timeThe accelerationmeasures10-—-9dhow quickly thebodypicksup and loses speed.E0FIGURE 3.13-The velocity graph for Esample 2Slide 3- 45Slide 3- 48EXAMPLE4r (seconds)s (meters)Finure 3.14 shows the fiee fall ofa heaw ball1=06ro.160# 16r0mbearing released fiom rest at time t-0 sect=1OF5(a)How many meters does the ball fall in the frst 2-10sec?160(b)What is its velocity, speed, and acceleration15when t=2?256001=?20t=225= 16032h3016035(b)FIGURE 3.15 (a) The rock in Example 4.40(b) The graphs of s and as functions of1=30-45time; s is largest when w ds/dt - 0Thegraphofsis northepath of therock:It is a plot of height versus time.The slope1=OEFGURE 3.14A ballbearingpiyfalling from rest (Example 3).(a)here as a straight line.Slide 3-48Stide 3-47
2016/11/15 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 43 Velocity tells how fast an object is moving and the direction of motion When the object is moving forward (s increasing), the velocity is positive. When the object is moving backward (s decreasing), the velocity is negative. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 44 Speed measures the rate of progress regardless of direction. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 45 The rate at which a body's velocity changes is the body's acceleration. The acceleration measures how quickly the body picks up and loses speed. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 46 A sudden change in acceleration is called a "jerk". When a ride in a car or a bus is jerk, it is not that the acceleration involved are necessarily large but that the changes in acceleration are abrupt. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 47 EXAMPLE 4 Figure 3.14 shows the free fall of a heavy ball bearing released from rest at time t=0 sec. (a)How many meters does the ball fall in the first 2 sec? (b)What is its velocity, speed, and acceleration when t=2? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 48
The marginalcostofproductiony (dollars)is the rate of change of costSlope =withrespectto level ofy=c(x)marginal costproduction, so it is dc/dx.c(x+h)-c(x)f3.DerivativesinEconomicsaveragecost01+hoftheadditionalh(tons/week)isof.steelprodtFIGURE3.16Weekly steel productionde=limx+h)-c()c(x) is the cost of producing x tons perdxweek.The cost of producing an additional"0htons isc(x+h)-c(x)CoiarginalcostCofproductionSlide3-49Slide 3-50Sometimes themarginal?costofproduction islooselydefined to be theextra costofproducingone unit:Acc(x+1)-c(x)c(x)Ax10/BF+CFIGURE 3.17_The marginal cost de/dx isapproximatelytheextracostcofFIGURE3.18The graphs for Exercise 22.producing Ax I more unit.Slide3-51Slide 3-52HRULESDerivative Product RuleIf and uaredifferentiable atx,then so is theirproduct v,and）3.3EXAMPLE1Find the derivatives of the finctions:(a) y=(r +1(x +3)Derivatives ofProducts,Quotient, and Negative(x+(b) y=1Powers山Slide 3-54
2016/11/15 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 3 - 49 3. Derivatives in Economics Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 50 The marginal cost of production is the rate of change of cost with respect to level of production, so it is dc/dx. average cost of each of the additional h tons of steel produced c x h c x ( ) ( ) h marginal cost of production 0 ( ) ( ) lim h dc c x h c x dx h Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 51 Sometimes the marginal cost of production is loosely defined to be the extra cost of producing one unit: ( 1) ( ) '( ) 1 c c x c x c x x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 52 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.3 Derivatives of Products, Quotient, and Negative Powers Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 54 EXAMPLE 1 Find the derivatives of the functions: 2 3 ( ) ( 1)( 3) a y x x 1 1 2 ( ) ( ) b y x x x
RULE6Derivative Quotient RuleRULE2Power Rule (General Version)If u and w are differentiable at x and if u(x) + O, then the quotient u/u is differ- entiable at x, andIf n is any real number, then+=e-!ah()-dfor all x where the powers x* and x are defined7-1EXAMPLE2(a) Find the derivatives of the finctionThechoiceofwhichrulesto/+Iuseinsolvingadifferentiation(x -1)(x 2x)(b) Choose approprite nkes to find the derinative of yproblemcanmakeadifferenceinx4(howmuchworkyouhavetodo)9slide3-55Slide 3-58EXAMPLE3Find an equation for the tangent to the curve y=x-t the point (1,3)ExercisesFind the derivatives of the functions in Exercises 17-40.3.417.y18. ±:3x+xx2-419. g(x) 20.0)-+0.52+1-2Derivatives of21 w=(01 +2)22. w= (2 7)(x + 5)Trigonometric FunctionsV-15x +23, fo) = -24,w=V5+12V1+x-4V+Vo25, . =26.7=2广CoSlide 3-s7The derivative of the sine function is the cosine function量 (ina) c0 x1.Derivatives of theBasicTrigonometric FunctionsThe derivative of the cosine function is the negative of the sine function:%(005) -sin xSlide 3-59Slide 3-60
2016/11/15 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 55 EXAMPLE 2 (a) Find the derivatives of the function 2 2 1 1 t y t (b) Choose appropriate rules to find the derivative of Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 56 The choice of which rules to use in solving a differentiation problem can make a difference in how much work you have to do. Slide 3 - 57 Exercises EXAMPLE 3 Find an equation for the tangent to the curve at the point (1,3). 2 y x x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.4 Derivatives of Trigonometric Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 3 - 59 1. Derivatives of the Basic Trigonometric Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 60