3.3 Rules for Differentiation Calculus. Drum Roll please [In a Deep Announcer Voice] And now the moment YOU VE ALL been waiting for

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1 . Rules or Dieretiatio Calculus. RULES FOR DIFFERENTIATION Drum Roll please [I a Deep Aoucer Voice] A ow the momet YOU VE ALL bee waitig or Rule #1 Derivative o a Costat Fuctio I c is a costat value, the [ c ] = 0 This shoul ot be too earth shatterig to ou, sice the slope o a costat uctio is alwas 0! Eample: Let ( ) = 5. Fi '( ). Rule # Power Rule I is a umber, the = 1, provie 1 eists. : I sectio. our book istiguishes betwee beig a positive iteger (rule ), beig a egative iteger (rule 7) a beig a ratioal umber (rule 9, sectio.7). The istictio is mae so that the ma prove each separate case i the book. However, the use o the power rule is uchage or all three ieret values o. Eample: Prove the Power Rule or positive iteger values o. 5 Eample: Let ( ) =. Fi '( ). Eample: Let ( ) =. Fi '( ). 1 Eample: Let ( ) =. Fi '( ). 5

2 . Rules or Dieretiatio Calculus Rule : The Costat Multiple Rule I u is a ieretiable uctio o a c is a costat, the [ cu] = c u. Eample: Let 7 = 5. Fi. Eample: Let g( ) = 5. Fi g ' ( ). Rule : The Sum a Dierece Rule I u a v are ieretiable uctios o, the wherever u a v are ieretiable u [ u± v] = ± v Eample: Let = Fi '. = 1 ( ) Eample: Let g( ) +. Fi g '( ). Eample: Fi the equatio o the taget lie to the uctio ( ) = 6+5 whe =. Eample: Fi all poits where the graph o = has a horizotal taget lie. Eample: Let h( ) = ( + 1)( 5). Fi h' ( ). V Eample: The volume o a cube with sies o legth s is give b V = s. Fi s whe s = cetimeters. 5

3 . Rules or Dieretiatio Calculus Usig Rule, we kow that the erivative o the sum o two uctios is the sum o the erivatives o the two uctios. This oes ot work or the prouct a quotiet o two uctios. To illustrate this, we look at the ollowig eample. Eample: Fi. Rule 5: The Prouct Rule I u a v are ieretiable uctios o, the This is also writte as v [ uv] = u + v u [ uv ] = uv ' + vu ' Proo: For polomial uctios it is ot alwas ecessar to use the prouct rule, however, with trigoometric, epoetial, logarithmic, a other uctios, it is a ecessar tool. 5 ) = +. Fi '( ) without usig the prouct rule irst, the usig the prouct rule. Eample: Let ( ) ( )( 55

4 . Rules or Dieretiatio Calculus = + 5 7). Fi. Eample: Let ( )( Eample: Fi the equatio o the taget lie to the graph o ( ) = ( + 1)( + ) at the poit (1, ). Rule 6: The Quotiet Rule I u a v are ieretiable uctios o, a v 0 u v v u u =. v v This is also writte as u vu' uv' =. v v Proo: 56

5 . Rules or Dieretiatio Calculus Looks GREAT! Does t it?! Well, luckil or ou, it ai t that ba. With thaks to Sow White a the Seve Dwars, i we replace u with hi a v with ho (hi or high up there i the umerator a ho or low ow there i the eomiator), a lettig D sta or the erivative o, the ormula becomes hi ho D ( hi) hi D ( ho) D = ho ho I wors, that is ho ee hi mius hi ee ho over ho ho. Now, i Sleep a Seez ca remember that, it shoul t be a problem or ou. ( ) Eample: Fi + 1 Eample: Fi Eample: For a, i ' ( ) give the ollowig iormatio: g( ) = g' ( ) = h( ) = 1 h' ( ) = a) ( ) = g( ) + h( ) b) ( ) = h( ) c) ( ) = g( ) h( ) g ( ) ) ( ) = h( ) 57

6 . Rules or Dieretiatio Calculus Seco a Higher Orer Derivatives The irst erivative o with respect to is eote ' or. The seco erivative o with respect to is eote '' or. The seco erivative is a eample o a higher orer erivative. We ca cotiue to take erivatives (as log as the eist) usig the ollowig otatio: First erivative ' '( ) ( ) Seco erivative Thir erivative Fourth erivative ( ) '' ''( ) ''' '''( ) ( ) ( ) ( ) ( ) ( ) th erivative ( ) ( ) ( ) ( ) Eample: Fi Eample: Let ( ) =. Fi ''( ). 1 Eample: I ( ) ( ) =, i () 5 ( ). Notecars rom Sectio.: Power Rule; Prouct Rule; Quotiet Rule 58

2.4 Product & Quotient Rules

2.4 Product & Quotient Rules 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