2.4 Product & Quotient Rules

Transcription

1 Notes:. Prouct & Quotiet Rules. Prouct & Quotiet Rules ( ) is the y-value geeratig machie. is the slope value geeratig machie. The INCORRECT Prouct Rule The erivative o a prouct o two uctios a g is the prouct o the erivatives o a g. g = g The CORRECT Prouct Rule The erivative o a prouct o two uctios a g is. g = g + g Take turs, oe erivative at a time per term. The umber o actors will equal the umber o terms. Multiplicatio a aitio are both commutative, so this oe is iicult to mess up, uless you o it icorrectly. Say it with me, with caece a covictio: prime g, plus g prime : Eample 1: Fi the erivative o g ( 5 ) = usig the prouct rule, the veriy usig the power rule. Eample : Evaluate the ollowig. (a) 5 cos = b. [ cos si ] = Page 1 o 6

2 Notes:. Prouct & Quotiet Rules The INCORRECT Quotiet Rule The erivative o a quotiet o two uctios a g is the quotiet o the erivatives o a g. =, g 0, g 0 g g The CORRECT Quotiet Rule The erivative o a quotiet o two uctios a g is g g =, g 0 g g Subtractio is ot commutative, so this oe is more importat to get straight. A easier way to remember it is to thik o the umerator as the HI uctio (it is up high, ater all), a the eomiator as the LO uctio (sice it s i the bottom.) To make it more alliterative, let s actually call the eomiator the HO uctio (it oes rhyme with LO. ) Fially, let mea the erivative o... The Quotiet Rule ow becomes The HOHI (Quotiet) Rule The erivative o a quotiet o two uctios HI a HO is HI HOHI HIHO, HO 0 = HO HO HO Thik o the Seve Dwars. Thik o a Dysleic Mr. Wilso greetig Tim. Thik o Sata Clause i a hurry. Eample : Fi the erivative o each o the ollowig: (a) y = 1 (b) si = cos Page o 6

3 Notes:. Prouct & Quotiet Rules Eample : Fi a equatio o the taget lie to the graph o 1/ = + 5 at = 1 Eample 5: Evaluate each o the ollowig. si (a) = (b) ( si ) = Derivatives o the other Trigoometric Fuctios [ ta ] = sec [ cot ] = csc [ sec ] = sec ta [ csc ] MEMORIZE THESE IF YOU HAVEN T ALREADY = csc cot Eample 6: Fi the erivatives o each o the ollowig usig your memory. (a) g = ta (b) y = sec Page o 6

4 Notes:. Prouct & Quotiet Rules Eample 7: Dieretiate each o the ollowig uctios, the show that they have equivalet erivatives. 1 cos (a) = (b) g = csc cot si Higher-Orer Derivatives Recall the ollowig relatios amog the ollowig three uctios. st ( ) positio uctio vt ( ) = s ( t) velocity uctio ( ) ( ) ( ) at = v t = s t acceleratio uctio The otatio s ( t) is calle the seco erivative o ( ) st a we ca rea it as s ouble prime o t. The seco erivative is a eample o a higher-orer erivative. Why stop at two??? First erivative: y Seco erivative: y Thir erivative: y y y y Fourth erivative: ( ) y ( ) ( ) y M th erivative: ( ) y ( ) ( ) y Page o 6

5 Eample 8: (a) I y = 5 + 6, i Notes:. Prouct & Quotiet Rules ( 5 ) y ( ). (b) i = , ( 58 ) ( ). Eample 9: Evaluate: 7 (a) [ ] 7 si = (b) cos = Eample 10: ( ) g( ) g The table above gives values or two ieretiable uctios a their erivatives at selecte values o. Use the table to evaluate the ollowig. (a) h ( 0) i h = (b) h ( 1) i h = g g Page 5 o 6

6 Notes:. Prouct & Quotiet Rules Eample 11: True Story: The positio uctio or a allig object o (or ear, 6 eet high) the surace o the moo is give by h t ( ) =.66t + h 0, where ( ) secos a h 0 is the iitial height, i eet. ht is the height i eet at t (a) I you were eatig at a Italia restaurat o the moo (great oo, but o atmosphere), a your meatball rolle o the.-oot high table, ater how may secos woul it hit the loor? How ast was the meatball travellig whe it hit the grou? I you retrieve it withi 5 secos, woul you still eat it? (b) Usig the give equatio, etermie how may times stroger the acceleratio ue to gravity o Earth is tha that o the moo. (c) Base o your aswer to (b), etermie what your weight o the moo woul be? Diet problem solve? Page 6 o 6