EIEre{ Cx =101. polys. computation. ( xh ) Derivatives of. functions. fcxthl 3. I. exp. Q : what about der. Algebraically. C x.

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1 o o 3 Derivatives o pols and ep unctions dea : we want to speed up o derivatives b proving our computation ormulas Q : wat is te der C 3 Grapicall slope alwas is c Q? Algebraicall Lingo Ctl Lio So C C implies C O Q : wat about der XZ? also!? C EEre{ C2 2? C it GW 10 Constant t power Rules #

2 10 to or e Di O Lets o const n malt 7 2 X So We know! but wat about? GW Sum Rules E Find te der c 5 3 t X t t First convert to powers 5 3 e kt7 11 o Ten C k to GW # 2 # 3 Eponential Functions we know ow to ind 62 Wat about 2 e? Let C ind C 1 First lets tr to ind co GW # 4

3 Tis So we tink Co lim u 70 o L Cot a means Co nigo e n act Det e is deined to be te number stli?oe 2 Now C _ lio e + u T lim en u 20 Cd c# e! e GW 10 # 5

4 Product Quotient Rules You know X3 3e 3 2 So wat about? e e Eperiment : 3 4?? Y 43! at L ot equal 7 6 GW 11 Product Rule E Compute a 1 6 t e t e 431 etent 3 2 b [ L e 3 2 C ] et 5 G e 2 1 C i s %

5 tr 3e C Seki GW 11 # 2 prod rule ol tr to ind ow So know [ l 3 e ] ou wat about B GW 11 te Quotient Rule jingle Ee Compute / t4 a C3e T [? 3e ] T a ter 3e 2 let k Yz b gs#jc47k3e3ec47y47y4732ete3 e GW t # 34

6 g t adge ct g Proo at Product Rule an gc! nigo ltlgct lion ctlgct g e e 4t g C C g c u O eanl g CH g C t g C C D Proo at Quotient Rule Let c tg! Ten ad gcl C CPR ang g ca + g C C + g e g C e g! i gy CA s g C si lag g C C C e

7 12 Cos Siu Cos 33 Trig Derivatives Q : wat is sin? Our convention is tat is in radians Eperiment Gw # t # sin / * sin looks like cos! Aside : Special limits u ound sin 1*01 so sin liq > n o Co / Similarl cosl o o so n Fo C u o O Tus

8 12 12 > sin at Sin Sin so Teorem X im sing so And lion cos o Ten sin cos X rt sin Sin Ct lim u so lim into sin cos t Sin cos sink lim Sin cos 70 c Sin cos sin cos# sing! lim u o? cos X D Te CCos E Find t an? tan Zing ER costcos C o S sinsin z secz WS Der trig unctions WS # 2

9 13 o sin sin o 3 4 Cain Rule Q : can ou compute cos C? Q : wat about cos Cz? tis is about composition WS Cain Rule Some comments : g C C gc g C outside der der and plug inside inside back in inside o let u g e Ten cgci Cu s lu u E Find te derivatives Ca C cos u C cos ul u u 2 2 b gc g e 17 u ul 7 17 u 2e b u 17 eteiz

10 13 at na WS # 12 E Find e e e 7 34 e u e Te ace e u so e e A Te a > o rt d Le nca ten a n at na etra \ a na E arder Find der & ans an Has P tan s sj s t 5 u3 3? u 3 tan 3C ts s tan Cit 3 tan 3 t 5 sec tan > les s see C t 5 5? n 5

11 13 we Sca cgca W S # 3 dea o cain rule 1 Let C g e must sow i Ca C g ca g ca 2 Now O C Cat g 1 lim n a n o C g Cal 2 Linear approimation : te tangent line to C at an b is egetur Cb b t C b SO p C b b t C b wen is near b 3 Since g Cat is near gca a im C s Cat o C g Ca lion C gear g Cat 1 C goal 70 Lingo gca Scate Cgca g Ca pg

12 unction o T 3 5 mplicit Dierentiation mplicitl Deined Functions Te ez or 7 deine eplicitl as a unction o Te eq t 124 does not eplicitl deine a unction w? 2 i o is te input tere are 2 outputs : 412 However a unction at as t 24 does deine implicitl but it depends on wat part o te circle we use : Top > o Fi Bottom : YEO Et * XZ t 4 also deines as a unction at 1 Rigt : oo 5 Let : so T * However we usuall ocus on as a

13 line Similarl consider equations like XZ 4 Y is For anoter prett one consider sinc Coslz 3 working t Desnnos Me! wit equations instead at eplicit unctions allows us a lot more leibilit and it oten makes tings a lot tess mess Q : ow to we do calculus on equations? A : work implicitl E Suppose tat t 124 Find d and use tis to ind te tan to 4 4 at C we work implicitl C Tink but we do not solve or

14 o Fs line TC oka Steps to ind 0 Take der bot sides remembering C t 4 t 4 e Y 4 cain rule!! 2 t a 4 an 2 C C 2 Solve or 0 2 da 2 q s s! Now to ind te tan : point : C s! tir te line : t Fs t or 1 53

15 C cosc sin cos Ee Find c i sin C t cos 2 and * ll use prime [ Xt cos z t Cosc Sinai cosc Z inld Z collect co se s tz C 2 cos cosc tzsicz Cl sinatcoscit 2 Solve or!s Xcosc t2 sin r use it to ind d T least rt notation tis time 1 Der at bot sides remembering C tis dle* cos 2T it 2t Coslett 12t sin it WS 14 # 12

16 i nverse Trig Functions Recall : 1 Not all unctions ave an inverse! 2 as an inverse C ten i and C pictoriall : # C C is on o grap Y is on grap o 3 Be : in general e # i! E Grap and i it eists Ca ca Sin X o D WE b/c C does Not pass orizontal line test o C o O C it O o so T so i eists Co 0 it not a unction

17 b C sin 121 EXE relect over T 2 2 on grap at l a X 2 C on grap l l 1142 o bet arcsin sin is te incense o sin on Te EXE arc cos Cos cos O E X E T are un tan i tan Te L L % see book or oters sin / Csinj Lets ind sin sin [ since ] Cos sink ] cose 1 i C sine # T Sin so triangle work implicitl! simpli tis? : T FE cos i gives te

18 15 at WS Der nu Trig E Find te derivative at C Sec C u # 3 2 3

19 15 n 36 Derivatives at Logs Lets ind la! n c s e work implicitl e e can we simpli e et n X tis? So Qn n act tis can be etended a bit lnll s And # in a similar asion : toga a WS # 1 Logaritmic Dierentiation We know : X power unction 7? 7 C eponential? not power ume nor eponential

20 back an Ee Find! Logaritmic Di 1 Take n o bot sides simpli n n n X n X 2 mplicit Di la lu t t 4 tn 3 Sub in t n Ee Find & Y t tan t t L 3 1 n n [ Ct tan ] la 3 n t ta 2 T Y n Clt ta 3 + t 3 2 n Clt ta 3 4 3? t 3 2 n t tan t ta

21 Ee Eplain ow to proceed Ca T eponential b csin log c sin t power t cain d ln log

22 37 C Some Applications we just look at velocit and accel Recall : at gives te position o an object at time t ten position ; C t den vet der accel velocit : Ce Pos acceleration : Ce GW 16 # 2

23 17 cange tt 3 9 Related Rates E A large clindrical tank at radius Sm is being illed wit water into Te low o te water te tank is measured be 3 mymin Determine ow ast te dept at te water is canging 1 Picture 2 Rates & known unknown known : D } eigtrknown : Q1 wat d# we are 3 mymiu tring to ind? V is te volume rate at cange Cange 3 Relate Quantities c? wat? at? 2 V 25T it r 2 Relate Q2 wat strateg migt solve Rates t v t 25T tink v Vct we use to tis? As du t 251T D dt 5 Solve 325 it D d t dd +1 Tmi n z 38 Tunics WS All