Derivatives of Inverse Trig Functions
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1 Derivaives of Inverse Trig Fncions Ne we will look a he erivaives of he inverse rig fncions. The formlas may look complicae, b I hink yo will fin ha hey are no oo har o se. Yo will js have o be carefl o se he chain rle when fining erivaives of fncions wih embee fncions. I am going o go hrogh he erivaion of he formla for he erivaive of y = sin -. Yo o no have o se his meho when aking he erivaive, b i may help o see where he formla came from. The erivaion is no nearly as complicae as he formla may lea yo o believe. Then, I will o some eamples. Les I have some panicky people o here, I will give yo a char from his secion o se on he es. Yo o no have o memorize he formlas, b yo o have o know how o se hem correcly. The AP eam oes no, however, give yo he formlas. I sgges yo know how o ge hem from scrach an hen also be able o recognize ha yo are ealing wih inverse rig fncions. Wih ha knowlege, yo will be able o cope wih anyhing yo see abo his on he eam. Also, here will probably no be more han one of hese on he no-calclaor secion of he eam, so yo can easily skip he problem wiho really penalizing yor score, an if yo are on he calclaor allowe secion, yo will be able o se yor calclaor o help yo. Here is how o erive he formla for fining he erivaive, y/, of he inverse rig fncion: y = sin - If his fncion says ha "y is he angle whose sine eqals ", hen anoher way o wrie his same informaion is: = sin y " is eqal o he sine of y". = (cos y ) y/ Now we ake he erivaive, sing implici iffereniaion. /cos y = y/ Solve for y/. We wol like o rewrie his in erms of. We can o his by sing a reference riangle o fin he cosine of he angle y. y -^ y/ = / (sin - ) = /cos y = Remember ha we alreay know ha he sine of y is eqal o. From he riangle we see ha cos y, so we p ha ino he formla. This is or final answer. Or formlas le s skip he sep wih he reference riangle an go sraigh o he answer. However, for yor informaion, I o no have all of hese formlas memorize. I eiher o i from scrach if i is one I on' know, sing he reference riangle, or I se a able of inegraion. So o no feel ha o be a goo calcls sen yo nee o memorize hese. However, yo shol learn o recognize he paern as "he inegral of some sor of inverse rig fncion" an learn o se inegraion formlas; his laer is an essenial skill in calcls becase here are far oo many
2 formlas o know hem all (see he ables for inegrals a his sie hp:// if yo hink I am eaggeraing!). Here is a able wih all of he erivaives of inverse rig fncions. Noice ha for each of hese erivaives yo have o mliply by he erivaive of he embee fncion,. This is he chain rle - on' forge o se i! sin cos an Derivaives of Inverse Trig Fncions for < for < csc sec co for > for >
3 Derivaives of Inverse Trig Fncions - Eamples Here is he able wih all of he erivaives of inverse rig fncions. Noice ha for each of hese erivaives yo have o mliply by he erivaive of he embee fncion,. This is he chain rle - on' forge o se i! Derivaives of Inverse Trig Fncions sin for < csc cos for < sec an co for > for > Now for some eamples. A: y = cos - (/) = / = - y Noice ha here is an embee fncion of / or -. When aking he erivaive we will have o se he chain rle an mliply by he erivaive of ha fncion. Now we follow he formla from he able o ge he erivaive. The / or - goes ino he formla where he is. y Simplify some. This col be frher simplifie by fining a common enominaor ner he sqare roo an going from here, b yo on' have o o i. y y I will go hrogh ha era simplificaion once here so yo see he process. Yo migh have o know how o o i a some poin b yo o no have o o his simplificaion on he HW. I go a common enominaor an combine he erms ner he sqare roo. Then I facore o he / an ook is sqare roo. Tha / cancele wih he ha was osie he raical.
4 B: 3 y sin 3 3 y The embee fncion is = 3 - so when aking he erivaive we will have o se he chain rle an mliply by he erivaive of ha fncion. Now we follow he formla from he able o ge he erivaive. The 3/ or 3 - goes ino he formla where he is. y I am going o conine wih he simplificaions on his problem alhogh yo o no really nee o o so. I js wan yo o know how o o i in case yo ever nee o for a mliple choice qesion. y I go a common enominaor an combine he erms ner he sqare roo. Then I facore o he sqare roo of he /. y 6 9 I hink yo will be fine wih hese, js go slowly an be carefl. The possibiliies for algebraic errors abon!
5 Inverse Trig Fncions Derivaive Formlas sin cos an sec for for for
6 #: Fin y/ for he fncion y cos Homework Eamples #, 3, 5, 5 y cos y This is or fncion. This is he formla for he erivaive of he arccosine fncion: cos y In or fncion, = an / = #3: Fin y/ for he fncion y sin y y sin This is or fncion. This is he formla for he erivaive of he arcsine fncion: sin y In or fncion, = an / = #5: Fin y/s for he fncion y sec (s) y sec (s ) This is or fncion y y y y s s s s s s s s s s This is he formla for he erivaive of he arcsecan fncion: sec In or fncion, = s + an /s = If yo wan o o some more algebraic simplificaion, yo can. I is no reqire (an on he AP eam yo shol never ake he ime o o i), b yo shol be able o follow he algebra becase yo migh nee o o his a some poin. I se FOIL ner he raical sign an hen simplifie. Then I facore o a in he raical an simplifie.
7 #5: Fin y/ for he fncion y an csc, y an csc y / y Since we are given ha > 0, we know ha =. Ths, y ' = 0 This is or fncion. These are he formlas for he erivaive of he arcangen an arccosecan fncions: an ( ) where / csc / / Where = an / = #: Le f () = cos + 3. Fin he following. We are only going o o he secon wo pars. f () = cos + 3 f (0) = cos = f '() = - sin + 3 f '(0) = - sin = 3 Inverse fncion will incle he poin (, 0) since he fncion incles he poin (0, ) f - () = 0 f 3 becase f '(0) = 3 an he poin (0, ) eiss on he fncion. This is or fncion. They wan he vale of he fncion an he erivaive a = 0. Now hey wan he vale of he inverse fncion a =. If a poin (a, b) is on a fncion, he poin (b, a) eiss on he inverse (since o fin he inverse we reverse he an y). There is a hany relaionship beween he vale of he erivaive of a fncion an is inverse. The erivaive of he fncion evalae a a will be he reciprocal of he erivaive of he inverse evalae a f (a). Tha is he rick. The erivaives ms be evalae a a an f (a) respecively. Remember ha f (a) is js he y-vale of he poin a = a.
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