Robrto s Nots on Diffrntial Calculus Captr 5: Drivativs of transcndntal functions Sction 5 Drivativs of Trigonomtric functions Wat you nd to know alrady: Basic trigonomtric limits, t dfinition of drivativ, all basic diffrntiation rul. Wat you can larn r: How to diffrntiat functions involving trigonomtric functions. Non of t diffrntiation ruls w av dvlopd so far can b usd to comput t drivativs of trigonomtric functions. To do tat, w nd to go back to t dfinition of drivativ. Proof Tcnical fact y sin, tn y cos y cos, tn y sin I will sow you wy t first drivativ formula is corrct and lav t otr to you in t Larning qustions. By dfinition: d sin sin lim d 0 sin By using t addition formula for t sin function, tis bcoms: sin cos sin cos sin lim 0 Now w split t fraction in two pics by using t common factor of sin : sin cos sin sin cos lim lim 0 0 sin cos 1 sin cos lim lim 0 0 Sinc t limit is computd as approacs 0, t factors sin and cos, wic do not involv, can b movd out of t limit: cos 1 sin sin lim cos lim 0 0 But now w ar lft wit t two fundamntal trigonomtric limits, wos valus w know! Trfor, w can rac t conclusion: d sin sin 0 cos 1 cos d Diffrntial Calculus Captr 5: Drivativs of transcndntal functions Sction 5: Drivativs of trigonomtric functions Pag 1
Now tat t two basic formula ar in plac w can us tm, togtr wit otr suitabl diffrntiation ruls, to obtain t drivativs of t otr four basic trigonomtric functions. Proof y tan, tn Tcnical fact y sc y sc, tn y sc tan y csc, tn y csc cot y cot, tn y csc As bfor, I will sow you ow to prov t first formula and lav t otrs to you. By dfinition: sin tan cos W can trfor us t quotint rul and t trig drivativs w av just discovrd: sin cos sin cos tan cos cos cos cos sin sin cos sin cos By using t basic Pytagoran idntity w can conclud tat: d 1 tan sc d cos And now tat w av our si basic formula, w can combin tm wit all otr ruls and mtods w know alrady to diffrntiat any function tat involvs trigonomtric parts. I will sow you on ampl and giv you som mor in t Larning qustions for your plasur. Eampl: y tansin Lt s find t scond drivativ of tis function! For t first drivativ, w start by using t cain rul and t drivativ of t tangnt function: y sc sin sin Nt w us suitabl ruls, including tat for t drivativ of t sin function, to complt t computation: y sc sin cos For t scond drivativ w start wit t product rul and tn apply all suitabl ruls: y sc sin cos sc sin cos sc sin sc sin cos sc sin 9sin W nd two mor applications of t cain rul to complt t work: cos sc sin tan sin sc sin 9sin Tat looks long and ugly, but, tank goodnss, w ar don! Diffrntial Calculus Captr 5: Drivativs of transcndntal functions Sction 5: Drivativs of trigonomtric functions Pag
T drivativs of Summary y sin and y cos ar obtaind by using t dfinition of drivativ. T drivativs of y tan, y cot, y sc, y csc ar obtaind by using tir dfining idntitis and basic diffrntiation ruls. Common rrors to avoid It is asy to forgt t drivativs of trigonomtric functions, spcially t four dfind by fractions. Practic noug on tm so as to mmoriz tm ffctivly. Larning qustions for Sction D 5-5 Rviw qustions: 1. Eplain ow t ruls to diffrntiat basic trigonomtric functions ar obtaind. Mmory qustions: 1. Wat is t drivativ of f ( ) sin?. Wat is t drivativ of f ( ) cos? 4. Wat is t drivativ of f ( ) cot? 5. Wat is t drivativ of f ( ) sc?. Wat is t drivativ of f ( ) tan? Diffrntial Calculus Captr 5: Drivativs of transcndntal functions Sction 5: Drivativs of trigonomtric functions Pag
Computation qustions: Us appropriat ruls of diffrntiation to obtain t drivativs of t functions prsntd in qustions 1-48: 1... 4. 5. 6. 7. 8. 9. 10. 11. 1. y sin y cos y cos y 4 cos y tan y 10 sc y ( 5sin ) 4 4 y 4 sin 4 tan y ( 1) cos( 5) tan y y 6 cos( ) y cos 1. y sin( ) 14. 15. cos y cos y cos cos sin 16. y cos 17. 18. 19. 1 y tan cos y y tan sin cos sin 0. y cos 1.. y sin cos tan y sin log sin y cos ln. sin 4. y cos ln 5. y tan ln sin 6. y cos 1 ln tan y 7. sin 6 sin 8. y 9. y( 1) 0. y ln cos cos 1. ln y sin ln. y cos cos. y ln 4. y ( ) cos cos Diffrntial Calculus Captr 5: Drivativs of transcndntal functions Sction 5: Drivativs of trigonomtric functions Pag 4
5. y sin. 6. y sin 7. y sin 8. y 9. y cos ln cos 40. y ln sin 41. y cos sc cos 1 4. y ( 1) cos 4. y 6 44. y cos ln ln cos sin 45. y sin cos ln cos sin 46. y sin cos sin 47. y cos 1 ln 1 sin 48. y sin 1 cos cos ln 1 Comput t scond drivativ of t functions in qustions 49-54: 49. y ln sin cos 50. f ( ) sin 51. y ln sin 5. y ln cos 5. y sin ln 54. y cos ln 55. Dtrmin f 8 for f tan 56. Us implicit diffrntiation to dtrmin t scond drivativ of t function dfind by t quation y sin sin 1. sin 57. Dtrmin t 4 t drivativ of y 58. Comput t diffrntial of cos y at 0 wit d 1 6 59. Comput t drivativ of t function y cos 60. f is a diffrntiabl function, wat is t scond drivativ of cos g f? Diffrntial Calculus Captr 5: Drivativs of transcndntal functions Sction 5: Drivativs of trigonomtric functions Pag 5
Tory qustions: 1. Wic basic limit formula ar ndd in t proof of t drivativ formula for t sin function?. Wic diffrntiation rul is usd to obtain t formula for t drivativ of y tan? 4. Wic trig idntity is usd to prov t drivativ formula for t sin function? 5. T limit sin lim 0 tat is ndd to comput drivativs of trig functions is itslf a drivativ. Of wat function and at wat valu?. Wat is t 100 t drivativ of y sin? Proof qustions: 1. Us t dfinition of drivativ to obtain t drivativ of y cos. Us t dfinition of drivativ to prov tat sin cos 5. Sow tat lin tangnt to t curv orizontal. cos y at its y-intrcpt is sin. Us appropriat ruls of diffrntiation to obtain t drivativ of y sc and y csc d d 4. Prov tat sc sc tan witout using t quotint rul. 6. Comput t drivativ of t function y sin by using t dfinition of drivativ and cck tat tis is corrct by using appropriat diffrntiation ruls. Application qustions: Dtrmin t quation of t lin tangnt to ac of t curvs prsntd in qustions 1-1 at t givn point. Diffrntial Calculus Captr 5: Drivativs of transcndntal functions Sction 5: Drivativs of trigonomtric functions Pag 6
1. tan( y) at (0, 0).. y sin at,0? y tan sin at 4. y sin at, 4. 5. y cos 1 at 0, 6. y sin ln 4 1? at t y intrcpt. 7. sin y ln at,1 8. y tan sin at 9. y 10. cos 10 y sin y. tan( ) at =0. cos at, 0,1. 11. f sc tan at 1. f cos at t y-intrcpt. sin 1. Assuming, for simplicity, tat all monts to av 0 days, t amount of dayligt in Rd Dr is approimatly givn by t function: t 80 L 1 6sin 65 Comput ow muc ligt tim w can pct on Octobr 15 and ow fast it is canging. 14. An objct movs on t -ais, so tat aftr t sconds it is in position cos t. Wat is its acclration aftr t sconds? f tan and 15. Cck wtr t lins tangnts to g ln 5sin at 1 ar paralll. 16. An objct movs on t -ais, so tat aftr t sconds it is in position Wat is its acclration aftr t sconds? sint. Tmplatd qustions: 1. Construct a rasonably simpl function involving trigonomtric functions and comput its first and scond drivativs. Diffrntial Calculus Captr 5: Drivativs of transcndntal functions Sction 5: Drivativs of trigonomtric functions Pag 7
Wat qustions do you av for your instructor? Diffrntial Calculus Captr 5: Drivativs of transcndntal functions Sction 5: Drivativs of trigonomtric functions Pag 8