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Name: _________________ Period: ______ Date: ________________ AP Calc AB

Mr. Mellina

Chapter 4: More Derivatives

Sections:v 4.1 Chain Rule

v 4.4 Derivatives of Exponential and Logarithmic Functions v 4.2 Implicit Differentiation

v 5.6 Related Rates

HWSets

SetA(Section4.1)Page158,#’s1-31odd.

SetB(Section4.4)Page183,#’s1-20,37-40.

SetC(Section4.2)Page167#’s1-12,page169#’s59&60.

SetD(Section4.2)Page167,#’s27-30.APProblemsatendof4.2Section.

SetE(Section5.6)Page255,#’s9,11,13,17,19.

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4.1 The Chain Rule Topics

v Derivative of a Composite Function v “Outside-Inside” Rule

v Repeated Use of the Chain Rule v Power Chain Rule

Warm Up! Let𝑓 𝑥 = sin 𝑥,𝑔 𝑥 = 𝑥(,andℎ 𝑥 = 3𝑥 + 1.Writeasimplifiedexpressionforthecompositefunction.a. 𝑓(𝑔 𝑥 ) b. 𝑓 𝑔 ℎ 𝑥 c. 𝑓 ∘ ℎ 𝑥 d. 𝑓′ 𝑔(ℎ 𝑥 )

Derivative of a Composite Function We now know how to differentiate sin 𝑥 and 3𝑥 + 1, but how do we differentiate a composite like sin(3𝑥 + 1)? The answer is the ________ Rule, which is probably the most widely used differentiation rule in mathematics.

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Example1:ApplyingtheChainRuleFindthederivativea. 𝑓 𝑥 = 3𝑥( + 17𝑥 b. ℎ 𝑥 = 2

3425

c. 𝑓 𝑥 = 2𝑥( + 5 8 d. 𝑔 𝑥 = 𝑥( + 1 (9

The Chain Rule Formal Definition: If f is differentiable at the point 𝑢 = 𝑔(𝑥), and g is differentiable at x, then the composite function (𝑓 ∘ 𝑔)(𝑥) = 𝑓(𝑔(𝑥)) is differentiable at x, and

;;3𝑓<(𝑔(𝑥))= =

U-Substitution:

1. Let the “inside” function be ______.

2. Find ________ 3. Substitute the inside function with u and multiply by ;>

;3

4. Differentiate with respect to ____. 5. Substitute u back in.

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e. 𝑓 𝑥 = 𝑥5 − 2𝑥( + 5𝑥 𝑥( + 4 f. 𝑔 𝑥 = (3

3A4B9

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g. 𝑓 𝑡 = DE((D42

F

Example2:UsingtheChainRuleForeachofthefollowing,usethefactthat𝑔 5 = −3, 𝑔I 5 = 6, ℎ 5 = 3,andℎI 5 = −2tofind𝑓′(5),ifpossible.Ifitisnotpossible,statewhatadditionalinformationisrequired.a. 𝑓 𝑥 = 𝑔 𝑥 ℎ(𝑥) b. 𝑓 𝑥 = 𝑔 ℎ(𝑥) c. 𝑓 𝑥 = K 3

L(3) d. 𝑓 𝑥 = 𝑔 𝑥 5

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Example2:ChainRuleforTrigFunctionsFindthederivativeofthefollowinga. 𝑦 = 4𝑥 cos 𝑥 b. 𝑦 = cos 4𝑥c. 𝑓 𝑥 = cos( 4𝑥 d. 𝑔 𝑥 = cos 4𝑥(Example3:ApplyingthechainruleAnobjectmovesalongthex-axissothatitspositionatanytime𝑡 ≥ 0isgivenbythefollowingfunctions.Findthevelocityoftheobjectasafunctionoft.a. 𝑥 𝑡 = sin 𝑡( + 1 b. 𝑥 𝑡 = 𝑡 cos 𝜋 − 4𝑡

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Example4:AThree-LinkChainFindthederivativea. 𝑔 𝑡 = tan 5 − sin 2𝑡 b. For𝑦 = sinU 𝑥 ,find𝑦′|3YZ9

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4.4 Derivatives of Exponential & Logarithmic Functions

Topicsv Derivatives of 𝑒3 v Derivative of 𝑎3

v Derivative of the 𝑙𝑛𝑥 v Derivative of loga 𝑥

v Power Rule for Arbitrary Real Powers

Warm Up! SimplifytheExpressionusingPropertiesofExponentsandLogarithmsa. ln 𝑒bcd 3 b. log( 83EU c. ln 𝑥( − 4 − ln 𝑥 − 2 c. 3ln𝑥 − ln3𝑥 + ln 12𝑥( d. 𝑒3lna

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Example1:ChainRulewitheFindtheDerivative

a. 𝑓 𝑥 = 𝑒3 b. 𝑓 𝑥 = 𝑒53

c. 𝑓 𝑥 = 𝑥𝑒343A

d. 𝑓 𝑥 = 𝑒 3

e. 𝑓 𝑥 = 𝑒E3 + 𝑒3 5

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Example2:ChainRulewith𝒂𝒙FindtheDerivative

a. 𝑓 𝑥 = 33

b. 𝑓 𝑥 = 3B3

c. 𝑓 𝑥 = 3cot3

Derivative of 𝒂𝒙 For a >0 and 𝑎 ≠ 0, ;

;3(𝑎>)=

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Example3:ChainRulewithln𝒙FindtheDerivative

a. 𝑓 𝑥 = ln𝑥 b. 𝑓 𝑥 = ln 3𝑥 + 2

c. 𝑔 𝑥 = ln 𝑥5 + 𝑥 d. 𝑦 = ln𝑥 5

d. 𝑦 = 𝑒E3ln 𝑥

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4.2 Implicit Differentiation Topics

v Implicitly Defined Functions v Lenses, Tangents, and Normal Lines

v Derivatives of Higher Order v Rational Powers of Differentiable Functions

Warm Up! Solvefory’intermsofyandx.a. 𝑥(𝑦I − 2𝑥𝑦 = 4𝑥 − 𝑦 b. 𝑦I sin 𝑥 − 𝑥 cos 𝑥 = 𝑥𝑦I + 𝑦

Implicitly Defined Functions How do we find the slope when we cannot conveniently solve the equation to find the functions. For example the function 𝑥5 + 𝑦5 − 9𝑥𝑦 = 0 ? We treat y as a differentiable function of x and differentiate both sides of the equation with respect to ____, using the differentiation rules for sums, products, quotients, and the Chain Rule. Then solve for

in terms of x and y together to obtain a formula that calculates the

slope at any point (x, y) on the graph from the values of x and y. This process is called ______________ differentiation.

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Example1:DifferentiatingImplicitlyFind;j

;3.

a. 𝑦( = 𝑥 b. 𝑥5 + 𝑦5 = 18𝑥𝑦c. 𝑥( = 3Ej

34j d. 𝑥 + sin 𝑦 = 𝑥𝑦

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Example2:FindingSlopeofacurveusingImplicitDifferentiation Findtheslopeofthecurveatthegivenpoint.a. 𝑥( + 𝑦( = 25, 𝑎𝑡 3, −4 b. 𝑥 − 1 ( + 𝑦 − 1 ( = 25, 𝑎𝑡 3, 4

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Example3:TangentandNormaltoaFunction Findthelinesthataretangentandnormaltothecurveatthegivenpoint.a. 𝑥( − 3𝑥𝑦 + 2𝑦( = 5atthepoint 3, 2 b. 𝑥( cos( 𝑦 − sin 𝑦 = 0atthepoint 0, 𝜋

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Example5:FindingaSecondDerivativeImplicitlyFind;

Aj;3A

a. 2𝑥5 − 3𝑦( = 8

b. 𝑥A9 + 𝑦

A9 = 1

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c. 𝑥( + 𝑦( = 1 d. 𝑦( + 2𝑦 = 2𝑥 + 1

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4.2APHWProblemsAPCalculusAB2005TestQuestion5(FormB)Considerthecurvegivenby𝑦( = 2 + 𝑥𝑦.a. Showthat;j

;3= j

(jE3.

b. Findallpoints(x,y)onthecurvewherethelinetangenttothecurvehasslope2

(.

c. Showthattherearenopoints(x,y)onthecurvewherethelinetangenttothecurveis

horizontal.d. Letxandybefunctionsoftimetthatarerelatedbytheequation𝑦( = 2 + 𝑥𝑦.Attime

t=5,thevalueofyis3and;j;D= 6.Findthevalueof;3

;Dattimet=5.

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APCalculusAB2004TestQuestion4Considerthecurvegivenby𝑥( + 4𝑦( = 7 + 3𝑥𝑦.a. Showthat;j

;3= 5jE(3

kjE53.

b. ShowthatthereisapointPwithx-coordinate3atwhichthelinetangenttothecurveat

Pishorizontal.Findthey-coordinateofP.c. Findthevalueof;

Aj;3A

atthepointPfoundinpart(b).

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5.6 Related Rates Topics

v Related Rate Equations v Solution Strategy

v Simulating Related Motion

Warm Up! Findeachofthefollowingderivativesof𝑦 = 2𝑢 + 𝑝 − 𝑡.a. ;j

;> b. ;j

;D c. ;j

;3

Example1:RelatedRateEquationsSupposethataparticleP(x,y)ismovingalongacurveCintheplanesothatitscoordinatesxandyaredifferentiablefunctionoftimet.IfDisthedistancefromtheorigintoP,thenusingthechainrule,wecanfindanequationthatrelates;m

;D, ;3;D,and;j

;D.

a. 𝐷 = 𝑥( + 𝑦(,find;m

;D.

Any equation involving two or more variables that are differentiable functions of time t can be used to find an equation that relates their corresponding rates. These types of problems are called ___________ Rates. When one or more values in an equation change over time, we have related rates. We use related rates when the problem asks: How fast did something ___________?

If h is measured in cm and t is measured in minutes, then ;L;D

is measured in: ___________

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Example2:FindingRelatedRateEquationsUsethegiveninformationtowritetherelatedrateequation.a. AssumethattheradiusrofasphereisadifferentiableequationoftandletVbethe

volumeofthesphere.Findanequationthatrelates;o;Dand;p

;D.

b. Assumethattheradiusrandheighthofaconearedifferentiablefunctionsoftandlet

Vbethevolumeofthecone.Findanequationthatrelates;o;Dand;L

;D.

Strategy for Solving Related Rate Problems 1. ____________ the problem.

• In particular, identify the variable whose rate of change you seek and the variable (or variables) whose rate of change you know.

2. Develop a mathematical _________ of the problem. • Draw a picture (many of these problems involve geometric figures) and label the parts that are

important to the problem. Be sure to distinguish constant quantities from variables that change over time. Only constant quantities can be assigned numerical values at the start.

3. Write an ___________ relating the variable whose rate of change you seek with the variable(s) whose rate of change you know.

• The formula is often geometric, but it could come from a scientific application. 4. ______________ both sides of the equation implicitly with respect to time t. Be sure to follow all the

differentiation rules. The Chain Rule will be especially critical, as you will be differentiating with respect to the parameter t.

5. _____________ values for any quantities that depend on time. • Notice that it is only safe to do this after the differentiation step. Substituting too soon, “freezes

the picture” and makes changeable variables behave like constants, with zero derivatives. 6. _____________ the solutions.

• Translate your mathematical result into the problem setting (with appropriate units) and decide whether the result makes sense.

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Example3:SolvingRelatedRatesUsethegiveninformationtosolvetherelatedrateproblema. Ahot-airballoonrisingstraightupfromalevelfieldistrackedbyarangefinder500feet

fromthelift-offpoint.Atthemomenttherangefinder’selevationangleisqB,theangle

isincreasingattherateof0.14radiansperminute.Howfastistheballoonrisingatthatmoment?

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b. Apolicecruiser,approachingaright-angledintersectionfromthenorth,ischasingaspeedingcarthathasturnedthecornerandisnowmovingstraighteast.Whenthecruiseris0.6milesnorthoftheintersectionandthecaris0.8milestotheeast,thepolicedeterminewithradarthatthedistancebetweenthemandthecarisincreasingat20mpg.Ifthecruiserismovingat60mphattheinstantofmeasurement,whatisthespeedofthecar?

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c. Waterrunsintoaconicaltankattherateof9 rD9

stu.Thetankstandspointdownandhas

aheightof10ftandabaseradiusof5ft.Howfastisthewaterlevelrisingwhenthewateris6ftdeep?

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d. Joeyisperchedprecariouslyatthetopofa10footladderleaningagainstthebackwallofanapartmentbuilding(spyingonanenemyofhis)whenitstartstoslidedownthewallatarateof4feetperminute.Joey’saccomplice,Lou,isstandingontheground6feetawayfromthewall.HowfastisthebaseoftheladdermovingwhenithitsLou?

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Example4:MorePracticewithRelatedRatesUsethegiveninformationtosolvetherelatedrateproblema. Apebbleisdroppedintoapond,causingripplesintheformofconcentriccircles.The

radiusroftheoutercircleisincreasingatarateof1footpersecond.Whentheradiusis4feet,atwhatrateisthetotalareaAofthedisturbedwaterchanging?

b. Acone-shapedicicleisdrippingfromtheroof.Theradiusoftheicicleisdecreasingata

rateof0.2cm/hour,whilethelengthisincreasingatarateof0.8cm/hour.Iftheicicleiscurrently4cminradiusand20cmlong,isthevolumeoftheicicleincreasingordecreasing,andatwhatrate?

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c. A15-footladderisrestingagainstthewall.Thebottomisinitially10feetawayfromthewallandisbeingpushedtowardsthewallatarateof.25feet/sec.Howfastisthetopoftheladdermovingupthewall12secondsafterithasbeenpushed?