Cpter 3: Derivtives In tis cpter we will cover: 3 Te tnent line n te velocity problems Te erivtive t point n rtes o cne 3 Te erivtive s unction Dierentibility 3 Derivtives o constnt, power, polynomils n eponentil unctions 3 Te Prouct Rule n te Quotient Rule 33 Derivtives o trionometric unctions 34 Cin Rule Te erivtive o enerl eponentil unctions 36 Derivtives o loritmic unctions Loritmic ierentition 37 Cpter Review 3 Te tnent line n te velocity problems Te erivtive t point n rtes o cne A Te tnent problem: Consier te rp o unction, suc s te rp sown below: Fiure : Te rp o eneric unction n clcultion o te tnent line How cn we eine n in te tnent line to tis rp t point on te curve P,?
Te einition o te tnent line: We remember tt to in tnent line t te point P, we consier eneric point Q, on te rp, n clculte irst te slope o te secnt line PQ: m PQ, s sown in te iure Ten te tnent line t P, is te line wic psses trou P n wic s te slope: m T mpq, provie tt tis it eists QP Emple : Clculte te eqution o te tnent line to te rp o t, P, P, n P,4 Note: We sometimes reer to te slope o te tnent line to te rp o t point P, s te slope o te unction t P, Tis is becuse i we zoom in enou ner te point P, ten te unction i smoot t te point will pper to be line It is useul to see tt ormul cn lso be written s: m T, by mkin in te substitution Emple : Let unction Clculte te eqution o te tnent line t te point P, on te rp o tis B Te velocity problem: We ve seen in tt or n object wic moves lon strit line wit te position iven by s t, we ve tt te vere velocity o te object over time intervl t,t is iven by: 3 4 v t t t t cne in position ivie by te cne in time over tis time intervl n, Av t t eine te instntneous velocity or instntneous spee t v v, Av Emple 3: Do problem 3 rom te eercise set 7 t s were t is n rbitrry time t Equtions, n 4, wic re very similr, ive te instntneous rte o cne o te unction t P, Tis quntity is clle te erivtive o te unction t n is enote by '
Deinition : Te erivtive o unction t point in its omin D is iven by: 5 ', i tis it eists Optionl Note: In enerl, or unction y, te ierence quotient: 5 y rom to represents te vere reltive rte o cne o tis unction, wen cnes by Te reltive instntneous rte o cne o te quntity y t is lso te slope o te tnent line to te rp o Emple 4: Do emple 6 on pe 48 En Optionl Emple 5: t Fin te erivtive o te unction 4 9 t point ', wic is b Fin n eqution o te tnent line to te prbol 4 9 t P 3,6 Emple 5: Clculte,, - n or tese points n interpret tese s te instntneous rte o cne o t Note tt wen te erivtive is lre in mnitue t, ten te y vlues o te unction ner cne rpily, n wen te erivtive is close to, ten te curve is reltively lt ner, n te unction cnes very slowly ner 3
3 Te erivtive s unction Dierentibility Note tt in 3 we clculte te erivtive o unction t speciic point in its omin It is oten best to clculte i possible ' t ny point in its omin Tereore, eine: ' wen tis it eists Note: t We cn try to use lso ' s enerliztion o 5 rom 3 ltou te t t ormul bove is more nturl, n tereore lmost lwys use Given ny number, te epression sown in eines new unction ', clle te erivtive o Emple : I t Tis cn be interprete s te slope o te tnent line to te rp o t Te omin o smller tn te omin o is te set D ' eists 3, clculte ' Ten rw te rps o n ' on te sme es n conirm tt ' ives te slope o te tnent line to te rp o t ec point Emple : For, clculte ' Fin te omins o ' n o Grp bot unctions on teir respective omins Deinition : A unction is ierentible t i ' eists b A unction Emple 3: corresponin intervl is ierentible on, b or, or, or, i ' eists t ec point o te ' Were is ierentible n were it is not ierentible? Question: In enerl, ow cn unction il to be ierentible? First, let us estblis te ollowin importnt result: 4
Teorem : Consier unction : D R n D I is ierentible t, ten is continuous t n tke te it o tis equlity, rememberin tt Proo: Consier En o proo ' eists Tereore, one wy in wic unction is not ierentible t is wen te unction is iscontinuous t To unerstn best ow unction my il to be ierentible t, eine: ' l n 3 ' r Since ierentibility t mens tt ' eists, ten we cn estblis: Teorem : Consier unction : D R n D is ierentible t i n only i: ' l n ' r eist n i ' l ' r Tereore, unction cn il to be ierentible i one o te ollowin tkes plce: is iscontinuous t tink lso o te etreme cse in wic s te sme slope on eiter, or sie o te point but it is iscontinuous t, suc s :, wic is iscontinuous t, or Sow tt tis unction is not ierentible t, by clcultin ' n ' ' l or it oes not eist or ' or it oes not eist Note tt n ininite semi tnent r line mens tt te semi tnent line is verticl ' l n ' r bot eist but ' l ' r tis cse correspons to corner in te rp o See lso Fiure 7 on pe 59 in te tetbook or rpicl illustrtion o tese cses l r 5
3 Derivtives o constnt, power, polynomils n eponentil unctions From now on n or ew sections we inten to estblis ormuls or erivtives o elementry unctions lebric n trionometric, or trnscenentl unctions n or combintions o tese Tis will llow us to quickly clculte te erivtives o tese unctions, n tese clcultions will be use in mny pplictions s it ives te reltive instntneous rte o cne o tese epressions Finlly, we will collect ll tese ormuls in tble o erivtives, to ornize n elp us wit memorizin tese In tis section, we strt wit erivin ormuls or te erivtives o constnt, power, polynomil n eponentil unction U ormul rom 3, it is esy to see tt: c, or ny constnt c R Similrly:, n 3 3 Tis seems to suest te more enerl ormul: n n n, or n N, wic is esy to prove i we use te lebric ormul : n n n3 n n b b b b b n n b or te binomil ormul : n n n n! were n nk k k k k k! n k! Formul is clle te Power Rule or erivtives or s well n N We cn sow tt te power rule ols or n R First, ceck tt ols or n,, wic suests tt it ols or n Z proo will be iven in te net section Also ceck tt ols or n / n n / 3 Te ollowin enerl rules cn lso esily be veriie: 3 4 c c ', or ny constnt c R, n : ' ' n 5 ' ' by recllin tt n tt Formuls 3 to 5 llow us to eten ormuls n to oter epressions see emples 5 n 6 in te tetbook 6
Let us turn now to te erivtive o n eponentil unction, : R, ce First, u te lebric property:, we in tt :, tereore, we nee to etermine ' te enerl erivtive ' ' 5 ' ' ' ' Tere re mny possible equivlent einitions o te number e, wic is te bsis o te nturl loritmic unction suc s: 6 e n n Here, we eine e s te irrtionl number e,3 n suc tt e 7 note tt inormlly 7 is equivlent wit 6 7 mens tt e is te prticulr eponentil unction te unction e is see Fiure below or wic ' te slope o te tnent line t oriin or Fiure : Deinition o te number e U 5, tis prouces te useul ormul: 8 e e Emple : Do emple 8 n 9 in te tetbook 7
8 3 Te Prouct Rule n te Quotient Rule: Tese two enerl ormuls or erivtives te prouct rule n te quotient rule will llow us to clculte erivtives o combintions o unctions lernt in 3 Te enerl ormuls re: ' ' te prouct rule n ' ' te quotient rule Memorize tese ormuls correctly py specil ttention to te in ormul Proo o : ' ' Proo o : ' ' Note tt or we nee to keep trck o wic epression unction is te numertor n wic is te enomintor, wile or te orer o te terms o te prouct is not importnt Emple : Do Emples, 3, 4 n 5 in section 3 in te tetbook Memorize te ormuls sown t te en o te section 3
33 Derivtives o trionometric unctions Gol: Derive ormuls or ' cos cos ' tn tn ' cot cot ' Beore pplyin ormul in 3 to erive tese, note te rps o n o its erivtive built by mesurin te slope o te tnent lines to te rp o t ierent points below: Fiure : Te rps o n o its erivtive ' Tereore oo uess or U ormul in 3 : ' is ' cos Let us veriy tis lebriclly: cos cos cos cos Tereore, in orer to in ', we nee to prove ormul or wic we uesse s bein in cos n or For provin tt, consier te sector o unit circle sown in Fiure below: Fiure : A sector o n unit circle: note tt BC, OC cos, AO n AD tn Also, we ve tt: Are OBC Aresector OAB AreOAD cos tn cos cos 9
by u te squeeze teorem Tereore: Tereore, we ve sown tt: ' 3 u u rom te it lws, n n importnt it o its own Note tt implies more enerlly tt u, were u is ny epression wic pproces Emple : See Emple 5 in section 33 in te tetbook is n importnt result wic we will use to in cos In orer to in, note tt : cos cos cos Tereore: cos 4 wic implies te more enerl result: cos u 5 u u Returnin now to inin ormul or ', substitute n 4 in to in tt 6 cos, s uesse compre wit te rp in Fiure Similrly: cos cos cos cos cos cos cos cos ter u in n 4 Tereore: 7 cos 6 n 7 re importnt ormuls or te erivtives o n cos wic we will use oten
By u now te quotient rule ormul in 3, we cn esily erive tt eercise: 8 tn n tt: 9 cos cot A ormuls 6, 7, 8 n 9 to te tble o erivtives n memorize tis tble Emple : Emple in section 33 in te tetbook n eercises,,, 39 n 46 rom eercise set 33 34 Te Cin Rule: Gol: Derive n use rule or compose unction o te orm: u Motivtion: Until now, we know ow to clculte te erivtives o most o te simple lebric, trionometric n eponentil unctions However, wt i we nee to clculte te erivtive o compose unctions suc s: or Te cin rule erive in tis section will provie enerl ormul to clculte suc erivtives n oters Note tt: u u u u u u u u u u u u Formul is te cin rule, written sortly s: u u u n wic res : Te erivtive o unction pplie to noter unction is te u erivtive o te outer unction, evlute t te inner unction times te erivtive o te inner unction Emple : Do Emple in section 34: Clculte ' or Do Emple, tt is clculte ' or U, we cn esily erive: ln e ln ln e ln ' e ln ln n or ' e Tereore: 3 ln, erive in 3, n in reement wit tis Tis is te enerliztion o ormul or
U te cin rule, we esily enerlize ll ierentition ormul erive so r in te tble o ierentition, by tin n etr u or ec, or emple: n n n or n=,,3 becomes: 4 n u n n u u', or n=,,3, n so on or te oters Eercise: Generlize s suc ll ormuls in te tble o erivtives to obtin te erivtives o compose unctions Emple : Do Emples 3,4 n 8 in section 34 in te tetbook 36 Derivtives o loritmic unctions Loritmic Dierentition In tis section we will complete te tble o erivtives wit ormuls or te erivtives o te loritmic lo n ln unctions: We will lso lern tecnique clle loritmic ierentition wic llows us to clculte reltively quickly te erivtives o complicte epressions, typiclly involvin proucts, rctions n / or powers n eponents I Te erivtives o lo n ln First we sow tt: : lo ln note tt in we nee : n Wy? y To sow, recll tt y lo u Use implicit ierentition in n recll tt see 4: ln u' y ln y' y' rom, wic sows y ln ln Consier now e in to in tt: 3 ln u Tereore: Formuls n 3 ive te erivtives o loritmic unctions n tey cn be enerlize or composite unctions s: 4 u lo u' n 5 u ln u ln u' u
Te simplicity o ormuls 3 n 5 mkes te nturl loritm n te nturl eponentil very useul in Clculus n in oter sciences A ormuls, 3, 4 n 5 to te tble o erivtives n memorize ll ormuls in te tble o erivtives Emple pe 8: Fin ' Emple : pe 9: Fin ' y i ln 3 y y i y ln Emple 3: pe 9: Fin y ' i y ln Emple 4: pe 9: Fin ' y i y lo II Loritmic ierentition: To clculte reltively quickly te erivtives o more complicte unctions involvin proucts, quotients, powers n/or eponents, we cn pply te nturl loritm to bot sies o te eqution irst Tis tecnique is clle loritmic ierentition Note tt tere re some cses o te orm y, y'? clculte witout loritmic ierentition, wen y cnnot be Emple 7: pe 9: Clculte y ' i y 3/ 4 3 5 Emple 8 problem 43 rom Eercise set 6: Clculte y ' i y 37 Cpter Review In tis cpter, we : Lerne te einition o te erivtive o unction t point: ', n its interprettion n use in terms o te slope o te tnent line m, instntneous velocity v n instntneous rte o cne o T y unction: Section 3 Lerne te einition o ', its interprettion s unction, n te notion o ierentible unctions Cses o non-ierentibility n te rpicl menin o ierentible n non-ierentible unctions Section 3 Derive n lerne memorize ormuls or te erivtives o, e,, polynomils,, c,,,, trionometric n loritmic unctions n ornize tese in tble o erivtives Sections 3, 3, 33 n 34 n 36; 3
Lerne ow to use loritmic ierentition Section 36 Lern te min results einitions, teorems n ormuls rom tese sections best write review uie to inclue tese, ten solve te review problems rom pes 63 to 67 in te tetbook, incluin te Concept Ceck, True/ Flse n Eercises types o problems 4