Chapr Th Drivaiv Businss Calculus 99 Scion 5: Drivaivs of Formulas In his scion, w ll g h rivaiv ruls ha will l us fin formulas for rivaivs whn our funcion coms o us as a formula. This is a vry algbraic scion, an you shoul g los of pracic. Whn you ll somon you hav sui calculus, his is h on skill hy will pc you o hav. Thr s no a lo of p maning hr hs ar sricly algbraic ruls. Builing Blocks Ths ar h simpls ruls ruls for h basic funcions. W won prov hs ruls; w ll jus us hm. Bu firs, l s look a a fw so ha w can s hy mak sns. Eampl Fin h rivaiv of y f ( ) m + b This is a linar funcion, so is graph is is own angn lin! Th slop of h angn lin, h rivaiv, is h slop of h lin: f '( ) m Eampl Rul: Th rivaiv of a linar funcion is is slop Fin h rivaiv of f ( ) 5. Think abou his on graphically, oo. Th graph of f() is a horizonal lin. So is slop is zro. f ' 0 Eampl Rul: Th rivaiv of a consan is zro f Fin h rivaiv of This qusion is challnging using limis. W will show you h long way o o i, hn giv you a shorhan rul o bypass all his. f ( + h) f ( ) Rcall h formal finiion of h rivaiv: f '( ) lim. h 0 h Using our funcion f ( ), f ( + h) ( + h) + h + h. Thn f f '( ) lim h 0 h + h lim h 0 h ( + h) f ( ) h h lim h 0 ( + h) h lim h 0 lim h 0 f From all ha, w fin h + h + h h ( + h) This chapr is (c) 0. I was rmi by Davi Lippman from Shana Calaway's rmi of Conmporary Calculus by Dal Hoffman. I is licns unr h Craiv Commons Aribuion licns.
Chapr Th Drivaiv Businss Calculus 00 Luckily, hr is a hany rul w us o skip using h limi: Eampl n n Powr Rul: Th rivaiv f ( ) is f ( ) n g. Fin h rivaiv of Using h powr rul, w know ha if f ( ), hn f ( ). Noic ha g is ims h funcion f. Think abou wha his chang mans o h graph of g i s now ims as all as g f f h graph of f. If w fin h slop of a scan lin, i will b ; ach slop will b ims h slop of h scan lin on h f graph. This propry will hol for h slops of angn lins, oo: ( ) ( ) Rul: Consans com along for h ri; ( kf ) kf ' Hr ar all h basic ruls in on plac.
Chapr Th Drivaiv Businss Calculus 0 Drivaiv Ruls: Builing Blocks In wha follows, f an g ar iffrniabl funcions of. (a) Consan Mulipl Rul: ( kf ) kf ' (b) Sum (or Diffrnc) Rul: ( f + g) f ' + g' (or ( f g) f ' g' n n n (c) Powr Rul: Spcial cass: ( k) 0 ( ) () Eponnial Funcions: (bcaus (bcaus ( a ) ln a a 0 k k ) ) ) ln () Naural Logarihm: ( ) Th sum, iffrnc, an consan mulipl rul combin wih h powr rul allow us o asily fin h rivaiv of any polynomial. Eampl 5 p + + 0 8 Fin h rivaiv of 7.8 00 7 0 8 ( 7 +.8 + 00) 7 0 70 0 8 ( 7 ) + ( ) (.8) + ( 00) 0 8 ( ) + ( ).8 ( ) + ( 00) 9 7 ( ) + ( 8 ).8 + 0 9 + 0 7.8
Chapr Th Drivaiv Businss Calculus 0 You on hav o show vry singl sp. Do b carful whn you r firs working wih h ruls, bu pry soon you ll b abl o jus wri own h rivaiv ircly: Eampl 6 7 + Fin Wriing ou h ruls, w' wri 7 + 7() () + 0 Onc you'r familiar wih h ruls, you can, in your ha, muliply h ims h 7 an h ims, an jus wri ( 7 + ) Th powr rul works vn if h powr is ngaiv or a fracion. In orr o apply i, firs ransla all roos an basic raional prssions ino ponns: Eampl 7 Fin h rivaiv of y 5 + Firs sp ransla ino ponns: y 5 / + + 5 Now you can ak h rivaiv: + 5 / / 5 + 5 ( + 5 ) / 5 + 6 + 5. If hr is a rason o, you can rwri h answr wih raicals an posiiv ponns: 6 / 5 + 6 + 5 + + 5 5 B carful whn fining h rivaivs wih ngaiv ponns.
Chapr Th Drivaiv Businss Calculus 0 Eampl 8 Fin h quaion of h lin angn o g 0 whn. Th slop of h angn lin is h valu of h rivaiv. W can compu g. To fin h slop of h angn lin whn, valua h rivaiv a ha poin. g (). Th slop of h angn lin is -. To fin h quaion of h angn lin, w also n a poin on h angn lin. Sinc h angn lin ouchs h original funcion a, w can fin h poin by valuaing h original funcion: g () 0 6. Th angn lin mus pass hrough h poin (, 6). Using h poin-slop quaion of a lin, h angn lin will hav quaion y 6 ( ). Simplifying o slop-inrcp form, h quaion is y +. Graphing, w can vrify his lin is in angn o h curv. Eampl 9 Th cos o prouc ims is hunr ollars. (a) Wha is h cos for proucing 00 ims? 0 ims? Wha is cos of h 0 s im? (b) For f(), calcula f '() an valua f ' a 00. How os f '(00) compar wih h las answr in par (a)? (a) Pu f() / hunr ollars, h cos for ims. Thn f(00) $000 an f(0) $00.99, so i coss $.99 for ha 0 s im. Using his finiion, h marginal cos is $.99. (b) f ( ) / so f ( 00) hunr ollars $5.00. 00 0 No how clos hs answrs ar! This shows (again) why i s OK ha w us boh finiions for marginal cos.
Chapr Th Drivaiv Businss Calculus 0 Prouc an Quoin Ruls Th basic ruls will l us ackl simpl funcions. Bu wha happns if w n h rivaiv of a combinaion of hs funcions? Eampl 0 Fin h rivaiv of g ( ) ( )( + ) This funcion is no a simpl sum or iffrnc of polynomials. I s a prouc of polynomials. W can simply muliply i ou o fin is rivaiv: g( ) ( )( + ) + g' 6 + 6 Now suppos w wan o fin h rivaiv of 5 7 5 f +.5 7.5 + 0 + This funcion is no a simpl sum or iffrnc of polynomials. I s a prouc of polynomials. W coul simply muliply i ou o fin is rivaiv as bfor who wans o volunr? Noboy? W ll n a rul for fining h rivaiv of a prouc so w on hav o muliply vryhing ou. I woul b gra if w can jus ak h rivaivs of h facors an muliply hm, bu unforunaly ha won giv h righ answr. o s ha, consir fining rivaiv of ( g )( + ). W alray work ou h rivaiv. I s g '( ) 6 + 6. Wha if w ry iffrniaing h facors an muliplying hm? W g ( ), which is oally iffrn from h corrc answr. Th ruls for fining rivaivs of proucs an quoins ar a lil complica, bu hy sav us h much mor complica algbra w migh fac if w wr o ry o muliply hings ou. Thy also l us al wih proucs whr h facors ar no polynomials. W can us hs ruls, oghr wih h basic ruls, o fin rivaivs of many complica looking funcions.
Chapr Th Drivaiv Businss Calculus 05 Drivaiv Ruls: Prouc an Quoin Ruls In wha follows, f an g ar iffrniabl funcions of. (f) Prouc Rul: fg f ' g + fg' Th rivaiv of h firs facor ims h scon lf alon, plus h firs lf alon ims h rivaiv of h scon. Th prouc rul can n o a prouc of svral funcions; h parn coninus ak h rivaiv of ach facor in urn, mulipli by all h ohr facors lf alon, an a hm up. (g) Quoin Rul: f f ' g g g fg' Th numraor of h rsul rsmbls h prouc rul, bu hr is a minus insa of a plus; h minus sign gos wih h g. Th nominaor is simply h squar of h original nominaor no rivaivs hr. Eampl Fin h rivaiv of F ln This is a prouc, so w n o us h prouc rul. I lik o pu own mpy parnhss o rmin myslf of h parn; ha way I on forg anyhing. + F ' Thn I fill in h parnhss h firs s gs h rivaiv of, h scon gs alon, h hir gs lf alon, an h fourh gs h rivaiv of ln. F ' ( )( ln) + ( ) ln + Noic ha his was on w couln hav on by muliplying ou. ln lf
Chapr Th Drivaiv Businss Calculus 06 Eampl + Fin h rivaiv of y + 6 This is a quoin, so w n o us h quoin rul. Again, you fin i hlpful o pu own h mpy parnhss as a mpla: y ' Thn fill in all h pics: + ln + 6 y' + 6 ( ) ( + )( 8 ) ( ) Now for goonss saks on ry o simplify ha! Rmmbr ha simpl pns on wha you will o n; in his cas, w wr ask o fin h rivaiv, an w v on ha. Plas STOP, unlss hr is a rason o simplify furhr. Chain Rul Thr is on mor yp of complica funcion ha w will wan o know how o iffrnia: composiion. Th Chain Rul will l us fin h rivaiv of a composiion. (This is h las rivaiv rul w will larn!) Eampl y + 5. Fin h rivaiv of This is no a simpl polynomial, so w can us h basic builing block ruls y. I is a prouc, so w coul wri i as y ( + 5) ( + 5)( + 5) an us h prouc rul. Or w coul muliply i ou an simply iffrnia h rsuling polynomial. I ll o i h scon way: 6 y ( + 5) 6 + 0 + 5 5 y' 6 + 80 + 50 y + 5. W coul wri i as a prouc wih 0 facors an us h prouc rul, or w coul muliply i ou. Bu I on wan o o ha, o you? Now suppos w wan o fin h rivaiv of 0 W n an asir way, a rul ha will hanl a composiion lik his. Th Chain Rul is a lil complica, bu i savs us h much mor complica algbra of muliplying somhing lik his ou. I will also hanl composiions whr i wouln b possibl o muliply i ou.
Chapr Th Drivaiv Businss Calculus 07 Th Chain Rul is h mos common plac for suns o mak misaks. Par of h rason is ha h noaion aks a lil ging us o. An par of h rason is ha suns ofn forg o us i whn hy shoul. Whn shoul you us h Chain Rul? Almos vry im you ak a rivaiv. Drivaiv Ruls: Chain Rul In wha follows, f an g ar iffrniabl funcions wih y f ( u) an u g( ) (h) Chain Rul (Libniz noaion): y y u u Noic ha h u s sm o cancl. This is on avanag of h Libniz noaion; i can rmin you of how h chain rul chains oghr. (h) Chain Rul (using prim noaion): f ' f ' u g' f ' g g' ( ) (h) Chain Rul (in wors): Th rivaiv of a composiion is h rivaiv of h ousi, wih h insi saying h sam, TIMES h rivaiv of wha s insi. I rci h vrsion in wors ach im I ak a rivaiv, spcially if h funcion is complica. Eampl y + 5. Fin h rivaiv of This is h sam on w i bfor by muliplying ou. This im, l s us h Chain Rul: Th insi funcion is wha appars insi h parnhss: + 5. Th ousi funcion is h firs hing w fin as w com in from h ousi i s h squar funcion, (insi). Th rivaiv of his ousi funcion is (*insi). Now using h chain rul, h rivaiv of our original funcion is: (*insi) TIMES h rivaiv of wha s insi (which is + 5 ): y y' ( + 5) ( + 5) ( + 5) If you muliply his ou, you g h sam answr w go bfor. Hurray! Algbra works!
Chapr Th Drivaiv Businss Calculus 08 Eampl 5 y + 5 Now w hav a way o hanl his on. I s h rivaiv of h ousi TIMES h rivaiv of wha s insi. Fin h rivaiv of 0 Th ousi funcion is (insi) 0, which has h rivaiv 0(insi) 9. y y' 0 ( + 5) 9 0( + 5) ( + 5) Eampl 6 Diffrnia +5. This isn a simpl ponnial funcion; i s a composiion. Typical calculaor or compur syna can hlp you s wha h insi funcion is hr. On a TI calculaor, for ampl, whn you push h ky, i opns up parnhss: ^ ( This lls you ha h insi of h ponnial funcion is h ponn. Hr, h insi is h ponn + 5. Now w can us h Chain Rul: W wan h rivaiv of h ousi TIMES h rivaiv of wha s insi. Th ousi is h o h somhing funcion, so is rivaiv is h sam hing. Th rivaiv of wha s insi is. So + 5 + 5 ( ) ( ) ( ) Eampl 7 Th abl givs valus for f, f ', g an g ' a a numbr of poins. Us hs valus o rmin ( f g )() an ( f g ) '() a an 0. f() g() f'() g'() ( f g )() ( g f )() - 0 0-0 - - 0 0 - ( f g )( ) f( g( ) ) f( ) 0 ( f g )(0) f( g(0) ) f( ). ( f g ) '( ) f '( g( ) ). g '( ) f '( ). (0) ()(0) 0 an ( f g ) '( 0 ) f '( g( 0 ) ). g '( 0 ) f '( ). ( ) ( )().
Chapr Th Drivaiv Businss Calculus 09 Drivaivs of Complica Funcions You r now ray o ak h rivaiv of som mighy complica funcions. Bu how o you ll wha rul applis firs? Com in from h ousi wha o you ncounr firs? Tha s h firs rul you n. Us h Prouc, Quoin, an Chain Ruls o pl off h layrs, on a a im, unil you r all h way insi. Eampl 8 Fin ln( 5 + 7) Coming in from h ousi, I s ha his is a prouc of wo (complica) funcions. So I ll n h Prouc Rul firs. I ll fill in h pics I know, an hn I can figur h rs as spara sps an subsiu in a h n: ( ln( 5 + 7) ) ( ) ln( 5 + 7) + ( ) ( ln( 5 + 7) ) Now as spara sps, I ll fin ( ) (using h Chain Rul) an 5 + 7 ( ln( 5 + 7) ) 5 (also using h Chain Rul). Finally, o subsiu hs in hir placs: ( ln( 5 + 7) ) ( ) ln( 5 + 7) + 5 5 + 7 (An plas on ry o simplify ha!) Eampl 9 Diffrnia z ( ) Don panic! As you com in from h ousi, wha s h firs hing you ncounr? I s ha h powr. Tha lls you ha his is a composiion, a (complica) funcion rais o h h powr. Sp On: Us h Chain Rul. Th rivaiv of h ousi TIMES h rivaiv of wha s insi.
Chapr Th Drivaiv Businss Calculus 0 z Now w r on sp insi, an w can concnra on jus h par. Now, as you com in from h ousi, h firs hing you ncounr is a quoin his is h quoin of wo (complica) funcions. Sp Two: Us h Quoin Rul. Th rivaiv of h numraor is sraighforwar, so w can jus calcula i. Th rivaiv of h nominaor is a bi rickir, so w'll lav i for now. 9 Now w v gon on mor sp insi, an w can concnra on jus h par. Now w hav a prouc. Sp Thr: Us h Prouc Rul: + An now w r all h way in no mor rivaivs o ak. Sp Four: Now i s jus a qusion of subsiuing back b carful now! +, so 9 +, so + 9 z. Phw!
Chapr Th Drivaiv Businss Calculus Wha if h Drivaiv Dosn Eis? A funcion is call iffrniabl a a poin if is rivaiv iss a ha poin. W v bn acing as if rivaivs is vrywhr for vry funcion. This is ru for mos of h funcions ha you will run ino in his class. Bu hr ar som common placs whr h rivaiv osn is. Rmmbr ha h rivaiv is h slop of h angn lin o h curv. Tha s wha o hink abou. Whr can a slop no is? If h angn lin is vrical, h rivaiv will no is. Eampl 0 Show ha f ( ) / is no iffrniabl a 0. / Fining h rivaiv, f ( ). A 0, his funcion is unfin. From h / graph, w can s ha h angn lin o his curv a 0 is vrical wih unfin slop, which is why h rivaiv os no is a 0. Whr can a angn lin no is? If hr is a sharp cornr (cusp) in h graph, h rivaiv will no is a ha poin bcaus hr is no wll-fin angn lin (a ring angn, if you will). If hr is a jump in h graph, h angn lin will b iffrn on ihr si an h rivaiv can is. Eampl Show ha f ( ) is no iffrniabl a 0. On h lf si of h graph, h slop of h lin is -. On h righ si of h graph, h slop is +. Thr is no wllfin angn lin a h sharp cornr a 0, so h funcion is no iffrniabl a ha poin.
Chapr Th Drivaiv Businss Calculus Erciss. Th graph of y f() is shown. (a) A which ingrs is f coninuous? (b) A which ingrs is f iffrniabl?. Th graph of y g() is shown. (a) A which ingrs is g coninuous? (b) A which ingrs is g iffrniabl?. Fill in h valus in h abl for ( f ( ) ), ( f ( ) + g ( ) ), an ( g( ) f ( ) ). f() f '() g() g '() ( f ( ) ) ( f ( ) + g ( ) ) ( g( ) f ( ) ) 0 0. Us h valus in h abl o fill in h rs of h abl. f() f '() g() g '() ( f ( ) g( ) ) f ( ) g( ) g f 0 0 Problms 5 an 6 rfr o h valus givn in his abl: f() g() f '() g '() ( f g )() ( f g )' () 0 0 0 0 5. Us h abl of valus o rmin ( f g )() an ( f g )' () a an. 6. Us h abl of valus o rmin ( f g )() an ( f g )' () a, an 0.
Chapr Th Drivaiv Businss Calculus 7. Us h informaion in h graph o plo h valus of h funcions f + g, f. g an f/g an hir rivaivs a, an. 8. Us h informaion in h graph o plo h valus of h funcions f, f g an g/f an hir rivaivs a, an. 9. Us h graphs o sima h valus of g(), g '(), (f g)(), f '( g() ), an ( f g ) '( ) a. 0. Us h graphs o sima h valus of g(), g '(), (f g)(), f '( g() ), an ( f g ) '( ) for.. Fin (a) D( ) (b). Fin (a) D( 9 ) (b) /. Calcula (( 5 )( + 7) ) ( 7 ) (c) D( ) () (c) D( ) () D( π ) by (a) using h prouc rul an (b) paning h prouc an hn iffrniaing. Vrify ha boh mhos giv h sam rsul.. If h prouc of f an g is a consan ( f() g() k for all ), hn how ar an ( g( ) ) g ( ) rla? ( f ( ) ) f ( ) 5. If h quoin of f an g is a consan ( rla? f g ( ) ( ) k for all ), hn how ar g. f ' an f. g '
Chapr Th Drivaiv Businss Calculus In problms 6, (a) calcula f '() an (b) rmin whn f '() 0. 6. f() 5 + 7. f() 5 0 + 7 8. f() + 9 + 6 9. f() + + 0. f() + + 7. f() + + 7.Drmin an. Fin (a) ( ) an (b) ( ). Fin (a) ( ), (b) ( ) 5 ( 5 + ). In problms 5 0, fin h rivaiv of ach funcion.. 5. f() ( 8) 5 6. f() (6 ) 0 7. f(). ( + 7) 5 8. f() ( + ) 6. ( ) 9. f() + 6 0. f() 5 ( + ). If f is a iffrniabl funcion, (a) how ar h graphs of y f() an y f() + k rla? (b) how ar h rivaivs of f() an f() + k rla?. Whr o f() 0 + an g() hav horizonal angn lins?. I aks T() hours o wav small rugs. Wha is h marginal proucion im o wav a rug? (B sur o inclu h unis wih your answr.). I coss C() ollars o prouc golf balls. Wha is h marginal proucion cos o mak a golf ball? Wha is h marginal proucion cos whn 5? whn 00? (Inclu unis.)
Chapr Th Drivaiv Businss Calculus 5 5. A manufacurr has rmin ha an mploy wih ays of proucion princ will b abl o prouc approimaly P() + 5( 0. ) ims pr ay. Graph P(). (a) Approimaly how many ims will a bginning mploy b abl o prouc ach ay? (b) How many ims will an princ mploy b abl o prouc ach ay? (c) Wha is h marginal proucion ra of an mploy wih 5 ays of princ? (Wha ar h unis of your answr, an wha os his answr man?) 6. An arrow sho sraigh up from groun lvl wih an iniial vlociy of 8 f pr scon will b a high h() 6 + 8 f a scons. (a) Drmin h vlociy of h arrow whn 0, an scons. (b) Wha is h vlociy of h arrow, v(), a any im? (c) A wha im will h vlociy of h arrow b 0? () Wha is h gras high h arrow rachs? () How long will h arrow b alof? (f) Us h answr for h vlociy in par (b) o rmin h acclraion, a() v '(), a any im. 7. If an arrow is sho sraigh up from groun lvl on h moon wih an iniial vlociy of 8 f pr scon, is high will b h().65 + 8 f a scons. Do pars (a) () of problm 0 using his nw quaion for h. 8. f() + A + B + C wih consans A, B an C. Can you fin coniions on h consans A, B an C which will guaran ha h graph of y f() has wo isinc "vrics"? (Hr a "vr" mans a plac whr h curv changs from incrasing o crasing or from crasing o incrasing.)