Chaper The Derivaive Applie Calculus 107 Secion 4: Prouc an Quoien Rules The basic rules will le us ackle simple funcions. Bu wha happens if we nee he erivaive of a combinaion of hese funcions? Eample 1 Fin he erivaive of h( = ( 4 11( + This funcion is no a simple sum or ifference of polynomials. I s a prouc of polynomials. We can simply muliply i ou o fin is erivaive: 4 h = 4 11 + = 4 11+ 1 ( ( ( '( = 16 11+ 6 h Now suppose we wane o fin he erivaive of 5 7 5 f = 4 + 1.5 11 7.5 + 10 + ( ( ( This funcion is no a simple sum or ifference of polynomials. I s a prouc of polynomials. We coul simply muliply i ou o fin is erivaive as before who wans o voluneer? Noboy? We ll nee a rule for fining he erivaive of a prouc so we on have o muliply everyhing ou. I woul be grea if we can jus ake he erivaives of he facors an muliply hem, bu unforunaely ha won give he righ answer. o see ha, consier fining erivaive of ( ( 4 g = 11( +. We alreay worke ou he erivaive. I s g '( = 16 11+ 6. Wha if we ry iffereniaing he facors an muliplying hem? We ge ( 1 ( 1 = 1, which is oally ifferen from he correc answer. The rules for fining erivaives of proucs an quoiens are a lile complicae, bu hey save us he much more complicae algebra we migh face if we were o ry o muliply hings ou. They also le us eal wih proucs where he facors are no polynomials. We can use hese rules, ogeher wih he basic rules, o fin erivaives of many complicae looking funcions. This chaper is (c 01. I was remie by Davi Lippman from Shana Calaway's remi of Conemporary Calculus by Dale Hoffman. I is license uner he Creaive Commons Aribuion license.
Chaper The Derivaive Applie Calculus 108 Derivaive Rules: Prouc an Quoien Rules In wha follows, f an g are iffereniable funcions of. (a Prouc Rule: fg = f ' g + ( fg' The erivaive of he firs facor imes he secon lef alone, plus he firs lef alone imes he erivaive of he secon. The prouc rule can een o a prouc of several funcions; he paern coninues ake he erivaive of each facor in urn, muliplie by all he oher facors lef alone, an a hem up. (b Quoien Rule: f f ' g = g g fg' The numeraor of he resul resembles he prouc rule, bu here is a minus insea of a plus; he minus sign goes wih he g. The enominaor is simply he square of he original enominaor no erivaives here. Eample Fin he erivaive of h( = ( 4 11( + This is he same funcion we foun he erivaive of in Eample 1, bu le's use he prouc rule an check o see if we ge he same answer. For his firs eample, we will provie a lo more eail an seps han one usually acually shows when working a problem like his. Noice we can hink of h( as he prouc of wo funcions Fining he erivaive of each of hese, f ( = 1 g ( = 1 f ( 4 11 = an g ( = +. Using he prouc rule, h = f g + f g = 1 + + 4 11 1 ( ( ( ( ( ( ( ( ( To check if his is equivalen o he answer we foun in Eample 1 we coul simplify: 1 1 h = 1 + + 4 11 1 = 1 + 6 + 4 11 = 16 + 6 11 ( ( ( ( ( From his, we can see he answers are equivalen.
Chaper The Derivaive Applie Calculus 109 Eample Fin he erivaive of F( = e ln This is a prouc, so we nee o use he prouc rule. I like o pu own empy parenheses o remin myself of he paern; ha way I on forge anyhing. ( = ( ( + ( ( F ' Then I fill in he parenheses he firs se ges he erivaive of e, he secon ges alone, he hir ges e lef alone, an he fourh ges he erivaive of ln. 1 e F '( = ( e ( ln + ( e = e ln + ln lef Noice ha his was one we couln have one by muliplying ou before aking he erivaive. Eample 4 4 + 4 Fin he erivaive of y = + 16 This is a quoien, so we nee o use he quoien rule. Again, you fin i helpful o pu own he empy parenheses as a emplae: ( ( ( ( y ' = ( Then fill in all he pieces: 4 + 4 ln 4 + 16 + 4 48 y ' = + 16 4 ( ( ( ( ( Now for gooness sakes on ry o simplify ha! Remember ha simple epens on wha you will o ne; in his case, we were aske o fin he erivaive, an we ve one ha. Please STOP, unless here is a reason o simplify furher. Eample 5 Suppose a large ank conains 8 kg of a chemical issolve in 50 liers of waer. If a ap is opene an waer is ae o he ank a a rae of 5 liers per minue, a wha rae is he concenraion of chemical in he ank changing afer 4 minues?
Chaper The Derivaive Applie Calculus 110 Firs we nee o se up a moel for he concenraion of chemical. The concenraion woul be measure as kg of chemical per lier of waer, kg L. The number of kg of chemical says consan a 8 kg, bu he quaniy of waer in he ank is increasing by 5 L/min. The oal volume of waer in he ank afer minues is 50 + 5, so he concenraion afer minues is 8 c ( = 50 + 5 To fin he rae a which he concenraion is changing, we nee he erivaive: ( 8 ( 50 + 5 ( 8 ( 50 + 5 c ( = 50 + 5 ( ( 0 ( 50 + 5 ( 8( 5 40 ( 50 + 5 ( 50 + 5 = = ( 50 + 5 4 A = 4, c (4 = 40 0.00816 kg / L Noe ha he unis here are kg per lier, per minue, or. In oher wors, his ells us ha min afer 4 minues, he concenraion of chemical is ecreasing by 0.00816 kg/l each minue. Reurning o our iscussion of business an economics opics, in aiion o oal cos an marginal cos, we ofen also wan o alk abou average cos or average revenue. TC( q The Average Cos (AC for q iems is he oal cos ivie by q, or AC( q =. You can q also alk abou he average fie cos, FC/q, or he average variable cos, TVC/q. The Average Revenue (AR for q iems is he oal revenue ivie by q, or TR/q. We alreay know ha we can fin average raes of change by fining slopes of secan lines. AC, AR, MC, an MR are all raes of change, an we can fin hem wih slopes, oo. AC(q is he slope of a iagonal line, from (0, 0 o (q, TC(q. AR(q is he slope of he line from (0, 0 o (q, TR(q.
Chaper The Derivaive Applie Calculus 111 slope = AC slope = AR Jus as we foun marginal Toal Cos, we can also fin marginal Average Cos. Eample 6 The cos, in housans of ollars, for proucing housan cellphone cases is given by C ( = + 0.004. Fin a The Fie coss b The Average Cos when 5 housan, 10 housan, or 0 housan cases are prouce c The Marginal Average Cos when 5 housan cases are prouce a The fie coss are he coss when no iems are prouce: C (0 = housan ollars b The average cos funcion is oal cos ivie by number of iems, so C ( + 0.004 AC( = = Noe he unis are housans of ollars per housans of iems, which simplifies o jus ollars per iem. + 5 0.004(5 A a proucion of 5 housan iems: AC(5 = = 5.8 ollars per iem 5 + 10 0.004(10 A a proucion of 10 housan iems: AC(10 = =.16ollars per iem 10 + 0 0.004(0 A a proucion of 0 housan iems: AC(0 = =.0 ollars per iem 0 Noice ha while he oal cos increases wih proucion, he average cos per iem ecreases, because he iniial fie coss are being isribue across more iems.
Chaper The Derivaive Applie Calculus 11 c For he marginal average cos, we nee o fin he erivaive of he average cos funcion. We can eiher calculae his using he quoien rule, or we coul use algebra o simplify he equaion firs: + 0.004 0.004 1 AC( = = + = + 1 0.004 = + 1 0.004. Taking he erivaive, AC = = ( 0.004 0.004 When 5 housan iems are prouce, AC (5 = 0.004 = 0.884 5 Since he unis on AC are ollars per iem, an he unis on are housans of iems, he unis on AC' ollars per iem per housans of iems. This ells us ha when 5 housan iems are prouce, he average cos per iem is ecreasing by $0.884 for each aiional housan iems prouce..4 Eercises 1. Use he values in he able o fill in he res of he able. f( f '( g( g '( ( f ( g( f ( ( g( g ( f 0 4 1 1 1 0 4 1. Use he informaion in he graph o plo he values of he funcions f + g, f. g an f/g an heir erivaives a = 1, an.. Use he informaion in he graph o plo he values of he funcions f, f g an g/f an heir erivaives a = 1, an. by (a using he prouc rule an (b epaning he prouc an hen iffereniaing. Verify ha boh mehos give he same resul. 4. Calculae (( 5 ( + 7
Chaper The Derivaive Applie Calculus 11 5. If he prouc of f an g is a consan ( f( g( = k for all, hen how are an ( g( g ( relae? ( f ( f ( 6. If he quoien of f an g is a consan ( relae? f g ( ( = k for all, hen how are g. f ' an f. g ' In problems 7 8, (a calculae f '(1 an (b eermine when f '( = 0. 7 7. f( = 8. f( = + 4 9. Deermine ( + 1( 7 an ( 5 + 1. 10. Fin (a ( e an (b ( e. 11. Fin (a ( e, (b ( e 5 1. A manufacurer has eermine ha an employee wih ays of proucion eperience will be able o prouce approimaely P( = + 15( 1 e 0. iems per ay. Graph P(. (a Approimaely how many iems will a beginning employee be able o prouce each ay? (b How many iems will an eperience employee be able o prouce each ay? (c Wha is he marginal proucion rae of an employee wih 5 ays of eperience? (Wha are he unis of your answer, an wha oes his answer mean?