Definition 2.1 (The Derivative) (page 54) is a function. The derivative of a function f with respect to x, represented by. f ', is defined by
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1 Chapter DACS Lok 004/05 CHAPTER DIFFERENTIATION. THE GEOMETRICAL MEANING OF DIFFERENTIATION (page 54) Defiitio. (The Derivative) (page 54) Let f () is a fctio. The erivative of a fctio f with respect to, represete b f '( ) lim 0 provie the limit eists. f ( ) f ', is efie b f ( ) The process to get the erivative sig Defiitio. is calle the ifferetiatio b sig the first pricipal. Notatios for writig the erivative: ', f ' '( ), f '( ), f f, Geometr to illstrate the cocept of ifferetiatio (page 55) crve: f (); poits: P 0, 0 a Q,
2 Chapter DACS Lok 004/05 Graiet of the lie PQ, 0 m PQ 0 f ( ) f ( 0) m PQ eq.(.) f ( 0 ) f ( 0) the, m PQ Whe Q approaches P, will approaches 0, hece 0 will te to zero. Whe Q is close eogh to P, the straight lie joiig P a Q will become taget ot the crve at P. Therefore, the graiet of the taget at P is m Defiitio. (page 56) f ( 0 ) f ( 0) lim 0
3 Chapter DACS Lok 004/05 Steps that ivolve i obtaiig the erivatives of the fctios sig efiitio (ifferetiatio from the first priciple) Step Give f (). Write the epressio f ( ). Step Obtai the ifferece f ( ) f ( ). Step Simplif the epressio f ( ) f ( ). Step 4 Fi the limit f '( ) ha 0 f ( ) f ( ). Eample.(page 57): B sig the ifferetiatio from the first pricipal, fi the erivatives of the followig fctios: 9 (b) () Eercise: f ( ) 5
4 Chapter DACS Lok 004/05 Remarks (page 60-6): A fctio is ot ifferetiable whe its graph has (i) poits of iscotiit (ii) vertices (iii) vertical taget Theorem. The Relatio betwee Cotiit a If f is ifferetiable at Differetiable (page 64) c, the f is cotios at c. Remarks (page 64): A fctio which is cotios at a poit ma be ot ifferetiable at that poit.. DIFFERENTIATION OF SIMPLE ALGEBRAIC FUNCTION (page 65) Theorem. (Derivative for Costat Fctio) (page 65) If c, c is a costat, for all, the 0. 4
5 Chapter DACS Lok 004/05 Eample.5 (page 66): Differetiate the followig fctios with respect to 00 (b) Teorem. (The Derivative for Positive Power Iteger) If, a is a positive iteger, the Eample.6 (page 67): Fi the erivatives of the followig fctios. 5 (b) f ( ) 99. DIFFERENTIATION RULES (page 68) Theorem.4 (Differetiatio of Mltiples) Let c. If si a ifferetiable fctio of, a c is a costat, the c Eample.7 (page 68): Fi the erivatives of the followig fctios (b) 5
6 Chapter DACS Lok 004/05 Theorem.5 (Defferetiatio of Sms) (page 69) Let a v be ifferetiable fctios of. If v, the v Remarks: (Defferetiatio of Differeces) (page 69) If v the v Eample.8 &.9 (page 69-70): Differetiate the followig fctios with respect to 4 5 (b) ( ) Theorem.6 (Differetiatio of Procts) (page 70) Let a v are ifferetiable fctios with respect to. If v, the v v Eample.0 (page 7): Differetiate with respect to if (4 )( ) Remarks (page 7): I some cases, it is easier to epa the epressio first before fiig its erivative. However, i cases whe epasio is ot possible or practical, the rle o ifferetiatio of procts will be eee. 6
7 Chapter DACS Lok 004/05 Theorem.7 (Differetiatio of Qotiet) (page 7) Let a v 0 be ifferetiable fctios with respect to. If, the v v v v Eample. &. (page 7 & 74): Differetiate the followig fctios with respect to 7 (b) 5 Theorem.8 (The Power Rle for All Iteger) (page 75) If, a is a iteger, the Notes: Theorem. is for positive power iteger ol while Theorem.8 is a etesio for all iteger. For 0, (page 75) 0 0 Eample. (page 75): Differetiate the followig fctios with respect to 9 f ( ) (b) 7
8 Chapter DACS Lok 004/05 Eercise: Fi the erivatives of the followig fctios. a, a is a costat (b) a, a is a costat (c) (f) 5 f ( ) () 0.5 (e) f ( ) f ( ) (g) ( )( ) 7 Smmar (page 76): The table below shows all the ifferetiatio rles where a v are ifferetiable fctios with respect to a c is a costat. f () f '( ) c v v v c v v v v v v.4 HIGHER ORDER DIFFERENTIATIONS Differetiatio of seco orer or higher. (page 78) 8
9 Chapter DACS Lok 004/05 f '( ) f ( ) First ifferetial coefficiet f ''( ) ( f ')' [ f ( ) f ( ) Seco ifferetial f '''( ) ( f '')' f f (4) ( ) ( f ''')' [ f ( )] coefficiet f ( ) [ f ( )] ( ) ( ) ( f ( ) )' [ f ( )] 4 4 f ( ) th ifferetial coefficiet (We se the prime smbol ol for ifferetiatio p to orer three. For erivative of higher orer we se itegers brackets to represet the orers of ifferetiatio.). If f (), the the seqece of ifferetiatios ca also be writte as, or ca be simplifie as ', '',,, ''', 4 4,,, (4),, (),. The otatio represets the vales of orer of ifferetiatio at certai vale 0 '( ), ), ' 0 (5) ( 0 0,
10 Chapter DACS Lok 004/05 Eample.4 (page 79): Show that satisfie the eqatio ''' '' ' 0. Eample.5 (page 79): If 5, fi the vales of (4) a (). Eercise at home: (Ttorial ) (page 65) Qiz A: No. a), b), a) (page 67) Qiz B: No., (page 76) Qiz C: No. a), c), e), a), c), e), a), b) (page 8) Qiz D: No. c), a), b), 4a) (page 4) Eercise : No. b), ), 6a).5 THE CHAIN RULE Theorem.9 (The Chai Rle) (page 8) If g is ifferetiable at a f is ifferetiable at the poit g (), the the composite fctio f g is ifferetiable at. I other wor, if f g() a g() the 0
11 Chapter DACS Lok 004/05 Two special formlas sig the chai rle whe is ifferetiable fctio with respect to. () '( ) / / (*) (**) Eample.9 (page 84): Fi for the followig fctios. a) 0 6 b) Sometime the chai rle ivolves more tha two fctios. For eample (b) (page 87 below) v v v v w w s s t t Eample. (page 88): Fi if 4. Eercise: 4. If f ( ) 4, fi its first five erivatives.
12 Chapter DACS Lok 004/05. Fi for the followig fctios. a) b) 5.6 DIFFERENTIATION FO TRIGONOMETRIC FUNCTIONS Differetiatio Formlas for the Trigoometric Fctios (page 90, 9, 97) [si ] cos [ cos ] si [ta ] sec [ cosec ] cosec cot [ sec ] sec ta [cot ] cosec Differetiatio Formlas for the Trigoometric Fctios whe g() is a fctio of (page 00) [si ] cos [ cos ] si [ta ] sec [ cosec ] cosec [ sec ] sec ta [cot ] cosec cot Eample. (page 9): (b) 5si Eample.4 (page 9): si( 5) (b) si ( 5)
13 Chapter DACS Lok 004/05 Eample.6 (page 95): cos () cos Eample.9 (page 97): 4 ta Eample. (page 99): Fi if (i) ta.7 DIFFERENTIATION OF LOGARITHMIC FUNCTION (page 0 & 04) If log a, the I geeral, if () a or log a e, log a log a, 0 e I the special case a e, we have log e l therefore log a, the b sig the chai rle we have log a log a Frther more whe e a e [l ], 0
14 Chapter DACS Lok 004/05 l e Sice l e [l ], 0 Eample. (page 04): log 5 Eample.4 (page 05): (b) l si cos (c) l () l si.8 DIFFERENTIATION OF EXPONENTIAL FUNCTION Epoetial fctios a a e Differetiatio Formlas for the Epoetial Fctios (page 09) a a l a e e Eample.7 (page 09): 0 (b) 5 0 Differetiatio Formlas for the Epoetial Fctios whe g() is a fctio of (page 09) a a l a ( ) e e Eample.8 (page ): 4
15 Chapter DACS Lok 004/05 e (c) e Eample.40 (page ): If e cos, fi (b).9 DIFFERENTIATION OF IMPLICIT FUNCTIONS I previos sectio we have iscsse the ifferetiatio of the fctio where ca be epresse i terms of, f (). This is call eplicit efiitio. However, ot all eqatios ca be eplicitl efie (where caot be epresse i terms of ol). Sch a relatio is sai to be implicit efie a we write i the form of F (, ) 0. The metho to obtai ifferetial coefficiets of fctios, which are implicitl efie, is calle implicit ifferetiatio. Theorem.0 (Differetiatio) If a are relate i a fctio, a eplicitl a implicitl efie, the 5
16 Chapter DACS Lok 004/05 Eample.4 (page 7): If 7, fi, a) b writte i terms of. b) b sig implicit ifferetiatio. Eample.4 (Page 8): If si, fi i terms of a. Eample.48 (page ): If Acos Bsi, where A a B are costats, prove that 4 0 Eample.49 (page 4): Fi for the fctio (b).0 DIFFERENTIATION OF PARAMETRIC FUCNTIONS (page 8) 6
17 Chapter DACS Lok 004/05 I some cases, implicit fctios ca be epresse i terms of parameters. The implicit relatioship of a ca be epresse i a simpler form b sig a thir variable kow as the parameter. For eample t a t t is a parametric eqatio of a crve with parameter t (page 9): The ifferetiatio of parametric fctio is a applicatio of the chai rle. t a also t t t Eample.5 (page 9): A crve is give b a parametric eqatio t a 4 4t 4t Fi b sig parametric ifferetiatio (b) i terms of a hece fi. Eample.55 (page 0): The parametric eqatios of a crve are give b 7
18 Chapter DACS Lok 004/05 t e a si t. Fi a i terms of t. Eercise: Fi for the give vale of t. t t t t 7 t 4 Eercise at home: Do Qiz E, F, G, H, I, J Eercise at home (Ttorial 4): (page 4-50) Eercise : Do o. 7a), 8a), 8), 8f), 9b), 9c), 9), 9e), 9i), 0a), 0b), 0c), 0e), 0f), a), e), g), h), k), l), m), c), f), a), b), c), e), i), o), 7b), 6b), 6c), 9a), 9b) 8
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