Gra (MCV4UE) AP Calculus Pag of 5 Drivativ of a Function & Diffrntiabilit Th Drivativ at a Point f ( a h) f ( a) Rcall, lim provis th slop of h0 h th tangnt to th graph f ( at th point a, f ( a), an th instantanous rat of chang of f ( with rspct to at = a. Sinc this limit plas a cntral rol in calculus, it is givn a nam an a concis notation. It is call th rivativ of f( at = a. It is not b f (a) an is ra as f prim of a. Othr Notation for Drivativs If f (, thn ' or ar us insta of f '( ). Libniz Notation lim rivativ of with rspct to 0 First Principls Dfinition of th Drivativ Th rivativ of f at th numbr a is givn b f '( a) lim h 0 f ( a h) h f ( a), provi that this limit ists. Eampl : Dtrmin a Drivativ using th First Principls Dfinition a) Us th first principls finition to trmin th rivativ of f (. b) What is th omains of f ( an f '(? c) What o ou notic about th natur of th rivativ? Dscrib th rlationship btwn th function an its rivativ. ) Dtrmin th valu of i) f '( 3), ii) f '( 0) an iii) f '( ) an what ar th valus rprsnt? ) Dtrmin th quation of th tangnt of th f ( at =.
Gra (MCV4UE) AP Calculus Pag of 5 Drivativ of a Function & Diffrntiabilit Eampl : Dtrmin a Drivativ using th First Principls Dfinition 3 a) Us th first principls finition to trmin th rivativ of f (. b) Dtrmin th quation of th tangnt lin at = - an show th lin on th givn cubic graph. Eampl 3: Dtrmin a Drivativ using th First Principls Dfinition Dtrmin th rivativ f '( of th function f (, 0. Eampl 4: Graphing f from f Graph th rivativ of th function f whos graph is givn. Working iagrams
Gra (MCV4UE) AP Calculus Pag 3 of 5 Drivativ of a Function & Diffrntiabilit Eampl 5: Dtrmin a Drivativ using th First Principls Dfinition a) Dtrmin th rivativ f '( of th function f (. b) Dtrmin an sktch th quation of th tangnt of f( at =. c) Dtrmin an sktch th quation of th lin that is prpnicular to th tangnt (normal) to f ( at = an that intrscts it at th point of tangnc. f ( Th Eistnc of Drivativs (Diffrntiabilit) A function f is sai to b iffrntiabl at a if f '( a) ists. At points whr f is not iffrntiabl, w sa that th rivativ os not ist. Thr common was for a rivativ to fail to ist ar shown. Cusp Th slops of th scant lins approach from on si an from th othr. / 3 Eg) f Vrtical Tangnt Th slops of th scant lins approach ithr or from both sis. Eg) f 3 Discontinuit Will caus on or both of th on-si rivativs to b nonistnt)
Gra (MCV4UE) AP Calculus Pag 4 of 5 Drivativ of a Function & Diffrntiabilit Eampl 6: Rcogniz an Vrif whr a Function is Non-iffrntiabl 5, if A picwis function f is fin b f (. Th graph 0.5, if of f consists of two lin sgmnts that form a vrt, or cornr, at (, 3). a) Us th first principls to prov that th rivativ f () os not ist. b) Graph th slop of th tangnt for ach on th function. How os this graph support our rsults in part a)?
Gra (MCV4UE) AP Calculus Pag 5 of 5 Drivativ of a Function & Diffrntiabilit Drivativs on a Calculator (ndriv) m f a h f a h h tangnt lin Th numrical rivativ of f as a function is not b NDER f( & ndriv (Ti Calculators) which is similar to th concpt of a 0. 000 lim a h or h0 Ti procurs: MATH;nDriv(prssion,variabl,valu) a 0.00 f a 0.00 f NDERf a 0.00 Lt h 0.00is mor than aquat. a-h a a+h Eampl 7: Computing th Numrical rivativs Comput th numrical rivativs 3,3, th numrical rivativ of a) NDER b) NDER,0, th numrical rivativ of at = 0 3 at = 3 Thorms If f has rivativ at = a, thn f is continuous at = a. If a an b ar an two points in an intrval on which f is iffrntiabl, thn f, taks on vr valu btwn a f ' b. f ' an Homwork: P. 05 #,3,5-3-6,8-0,, 4, 3,3 P. 4 #-6, 3-35, 39 Optional (Cal & Vctors) P. 73 #, 5, 6, 7, 8, 0,, 5, 9
Gra (MCV4UE) AP Calculus Pag of 4 Ruls for Diffrntiation Computing rivativs from th limit finition is tious an tim-consuming. In this sction, w will vlop som ruls that simplif th procss of iffrntiation. Constant Function Rul f ( c, c isa constant, f '( 0 n n n Powr Rul n n f (, n R, f '( n n n n Proof : (Constant Rul) f ( h) f ( f '( lim h0 h f ( c f ( h) c c c lim h0 h lim 0 h0 0 f ( 3 slop = 0 Eampl : Constant Rul Diffrntiat a) f ( 7, fin f '( b), fin Eampl : Powr Rul Diffrntiat using th powr rul a) 7 f ( b) f ( c) 0 3 Constant Multipl Rul f ( c, c isa constant, f '( cg'( n n n Eampl 3: Constant Multipl Rul Diffrntiat th following functions: a) 5 f ( 6 b) f ( 4 4 c) 8 g 3 5 3 ( ) 5
Gra (MCV4UE) AP Calculus Pag of 4 Ruls for Diffrntiation Sum Rul f ( p( q(,if p anq ar iffrntiabl, f '( p'( q'( p( q( p( q( Diffrnc Rul f ( p( q(, if p an q ar iffrntiabl, f '( p'( q'( p( q( p( q( Eampl 4: Sum an Diffrnc Rul Diffrntiat th following functions an simplif into positiv ponnt: a) f ( 4 7 6 b) 5 5 3 3 c) f ( 5 Prouct Rul: If p( f (, thn p' ( f '( f ( g'(. If u an v ar functions of, ( uv) u v u v Etn Prouct Rul p( f g h f ( h( p'( f '( f ( ' h( f ( f ( g'( h( f ( h'( h'( f '( h( f ( g'( h( f ( h'( f ' gh fg' h fgh' Not In som cass, it is asir to pan an simplif th prouct bfor iffrntiating, rathr than using th prouct rul. 5 7 5 6 4 f ( 3 (6 5) 8 5 f '( 6 75 Not Simplifing th rivativ mans to mak th rivativ into its factor form with positiv ponnts. Eampl 5: Using th Prouct Rul Us th Prouct rul to fin an simplif. 3 a) 53 b) f ( 3
Gra (MCV4UE) AP Calculus Pag 3 of 4 Ruls for Diffrntiation Quotint Rul: If both f an g ar iffrntiabl, thn so is th quotint. f ( If p (, 0 thn u v u v u v v f '( f ( g'( p' (. (Limits proof can b foun in th tt book) Eampl 6: Using th Quotint Rul 6 5 Us th Quotint rul to iffrntiat F ( an simplif. 3 4 Highr Drivativs Sinc th rivativ of a function f is itslf a function f ', w can tak its rivativ f ' '. Th rsult is a function call th scon rivativ of f an not b f ''. If f (, thn '' f ''( (Scon rivativ) 3 similarl, ''' f '''( 3 (Thir rivativ) n ( n) ( n) In gnral f ( n (n th rivativ) Eampl 7: Highr-Orr rivativs 4 3 Fin th first fiv rivativs of 5 3 6.
Gra (MCV4UE) AP Calculus Pag 4 of 4 Ruls for Diffrntiation Eampl 8: Fin th quation for a Tangnt 3 Dtrmin th slop of th tangnt to th curv 5 at =. Eampl 9: Application of Drivativs a) Dtrmin th point on th graph f ( 5 whr th tangnt(s) is/ar horizontal. b) Dtrmin th quation of th tangnt lin(s) to th function that passs through th point P(0.5, 7). c) Illustrat th tangnt lins an th function on th graph provi. Homwork: P. 4 #-4, 46,47 Cal & Vc (Optional) P. 8 # -4, 6, 7c, 8a, 9bf, 0-4, 7a,, 5a(iii) P. 90 # -f, 5a-c, 6, 7a, P. 97 # 4bcf, 5,6,7,9,0,,4,5
Gra (MCV4UE) AP Calculus Pag of 3 Vlocit an Othr Rats of Chang Motion on a straight Lin An objct that movs along a straight lin with its position trmin b a function of tim, s (t), has a vlocit of v( t) s'( t) an an acclration of a( t) v'( t) s''( t) at tim t. s v s In Libniz notation, v a t t t About Vlocit If v(t) > 0, objct is moving in a positiv irctions (right or up) at tim t; If v(t) < 0, objct is moving in a ngativ irctions (lft or own) at tim t; If v(t) = 0, objct is not moving an a chang in irction ma occur at tim t. About Acclration If a(t) > 0, objct s vlocit is incrasing; If a(t) < 0, objct s vlocit is crasing; If a(t) = 0, objct s vlocit is constant an sta. Objct is sping up (acclrating) if v(t) a(t) > 0. Objct is slowing own (clrating) if v(t) a(t) < 0. s s s v > 0, a > 0 Sping up v > 0, a < 0 Slowing own v < 0, a < 0 Sping up s v < 0, a > 0 Slowing own t t Eampl : Motion to Displacmnt an Vlocit 3 Th position of a particl moving on a lin is givn b th quation s ( t) t t 60t, t 0, whr t is masur in scons an s in mtrs. a) What is th vlocit aftr sc an aftr 6 sc? b) Whn is th particl at rst? c) Whn is th particl moving in th positiv irction/ngativ irction? ) Fin th total istanc travll b th particl uring th first 6 scons. t t
Gra (MCV4UE) AP Calculus Pag of 3 Vlocit an Othr Rats of Chang Eampl : Vlocit an Acclration 3 A particl movs accoring to th quation of motion s( t) t 9t 8t, whr t is masur in scons an s in mtrs. a) Whn is th acclration 0? b) Fin th isplacmnt an vlocit at that tim. c) At t = 4, is th objct sping up or slowing own at t = 4? Eampl 3: Analzing motion unr gravit nar surfac of Earth A basball is hit vrticall upwar. Th position function s (t), in mtrs, of th ball abov th groun is s ( t) 5t 30t, whr t is in scons. a) Dtrmin th maimum hight rach b th ball. b) Dtrmin th vlocit of th ball whn it is caught m abov th groun. c) Whn will th ball b 6 m about th groun?
Gra (MCV4UE) AP Calculus Pag 3 of 3 Vlocit an Othr Rats of Chang Marginal Cost, Marginal Rvnu an Marginal Profit In conomics an financ, marginal cost is th chang in total cost that ariss whn th quantit prouc changs b on unit. That is, it is th cost of proucing on mor unit of a goo. Marginal rvnu is th aitional rvnu that will b gnrat b incrasing prouct sals b unit. Marginal profit is th trm us to rfr to th iffrnc btwn th marginal cost an th marginal rvnu for proucing on aitional unit of prouction. Profit maimization rquirs that a firm prouc whr marginal rvnu quals marginal costs. Eampl 4: Marginal Cost an Marginal Rvnu c 3 6 5 ollars to prouc raiators whn 8 to 0 raiators ar prouc, Suppos it costs an that r 3 3 givs th ollar rvnu from slling raiators. Your shop currntl proucs 0 raiators a a. Fin th marginal cost an marginal rvnu. Motion unr th Influnc of Gravit (Galilo Formulas) Galilo iscovr that th hight s t an vlocit v t of an objct toss vrticall in th air s ar givn as functions of tim b th formulas: st s0 v0t gt, vt v0 gt t s 0 is th position at tim t = 0. Th constants 0 s 0 v is th vlocit at t 0. 0 v g is th acclration u to gravit on th surfac of th arth (ngativ bcaus th up irction is positiv), whr g 9.8m / s Eampl 5: Fining Initial Conitions A bullt is fir vrticall from an initial hight s 0 0. What initial vlocit v 0 is rquir for th bullt to rach a maimum hight of 3 km (3000m)? Homwork: P. 35 ##,3,4,8,9,,3-6,9-5,8,9 Cal & Vctors (Optional) P. 7 # 3, 4, 6bc, 8, 0,,, 4, 5, 7
Gra (MCV4UE) AP Calculus Pag of 4 Drivativs of Trigonomtric Functions Drivativs of Trigonomtric Functions (sin cos (cos sin (tan sc csc csc cot (sc sc tan (cot csc Eampl : Drivativ of cos using th first principl of rivativ cos Vrif sin using th first principl of rivativs. Rcall : f ' For f lim h 0 cos f h f h Eampl : Drivativ of Sin an Cosin functions Diffrntiat a) sin 4 b) cos(5 6) c) f sin 4 3 Eampl 3: Drivativ of Sin an Cosin functions using th rivativ ruls Diffrntiat sin a) cossin3 b) 3 cos4 c) cos3
Gra (MCV4UE) AP Calculus Pag of 4 Drivativs of Trigonomtric Functions Eampl 4: Drivativ of Sin an Cosin functions using th rivativ ruls Fin if f cos cos sin cos f cos cos Hint: Tr: 3sin cos Eampl 5: Equation of tangnt lin Fin th quation of th tangnt lin to sin at th point whr. cos 6 Eampl 6: Driving th rivativ of tan an csc tan a) Vrif sc. b) Vrif csc csc cot.
Gra (MCV4UE) AP Calculus Pag 3 of 4 Drivativs of Trigonomtric Functions Eampl 7: Drivativ of trigonomtric functions Diffrntiat a) f 4 3 tan b) f sin 3 cos 3 c) 3 cos tan Eampl 8: Drivativ of othr trigonomtric functions Diffrntiat 3 a) csc (3 ) f b) sc cot c) tan sc
Gra (MCV4UE) AP Calculus Pag 4 of 4 Drivativs of Trigonomtric Functions Eampl 9: Diffrntiating with a Paramtr Fin th lin tangnt to th right-han hprbola branch fin paramtricall b - sct, tant, t at th point, whr t 4 Eampl 0: Applications of Drivativ of Trigonomtric functions Fin th slop of th tangnt lin to tan(csc whn sin, in th intrval 0,. Homwork: P. 46 #-36, 4,43 P. 53 #-3 Cal & Vctors (Optional) P. 56 # - 5, 4 P. 60 # - 5, 8, 0,
Gra (MCV4UE) AP Calculus Pag of Th Drivativs of Composit Functions (Chain Rul) Chain Rul: If both f an g ar iffrntiabl, thn th composit function h f g h '( f ' g g'. ( has a rivativ givn b u u, provi that an ist. u u (Proof can b foun in th tt book) Rcall: Prouct Rul If p( f (, p' ( f '( f ( g'(. Rcall: Quotint Rul f ( p (, 0 f '( f ( g'( p' (. Eampl : Using th Chain Rul Us th Chain rul to iffrntiat an simplif if i) 4 5 7 ii) ( 3 3 h iii) 5 6 7 4 Eampl : Appling th Chain rul using Libniz notation 3 If u u, u, fin at 4.
Gra (MCV4UE) AP Calculus Pag of Th Drivativs of Composit Functions (Chain Rul) Eampl 3: Appling th Chain rul with othr rivativ ruls Diffrntiat i) s 6 4 ( ii) f ( 3 7 3 Eampl 4: Equation of Tangnt lin Fin th quation of th tangnt at th point (, ) for f ( 8 4 Homwork: AP: (Sav for Chain Rul ) Cal & Vctors P. 05 #bf, 4, 5, 7 7
Gra (MCV4UE) AP Calculus Pag of 3 Implicit Diffrntiation If th functions ar prssing on variabl plicitl in trms of anothr variabl; for ampl, or 4, in gnral, f (. But othr functions ar fin implicitl b a rlation btwn an such as 5. In this cas it is possibl to solv th quation for to gt two functions fin b th implicit quations ar f ( 5 an f an g ar th uppr an lowr smicircls of th circl 5. 5 an so 5. Th graphs of 5 f ( 5 5 Consir th problm of trmining th slop of th tangnt to th circl 5at th point (3, -4). Sinc this point lis on th lowr circl, w coul iffrntiat th function 5 an substitut = 3. An altrnativ, which avois having to solv for plicitl in trms of, is to us th mtho of implicit iffrntiation which involvs with th following two simpl stps. Stp: Diffrntiat both sis rspct to. LEFT RIGHT Eampl : Diffrntiations to th Rspct Variabl Diffrntiat th following functions rspct to th givn variabl a) 4 3 5, fin b) 4a 3 5a, fin a Stp : Solv for c) 4a 3 5, fin ) 4 3 5a, fin
Gra (MCV4UE) AP Calculus Pag of 3 Implicit Diffrntiation Eampl : Diffrntiat an Implicit rlation a) If 5, fin. b) Fin th quation of th tangnt lin to th circl 5 at th point = 3. Eampl 3: Using Implicit iffrntiation to trmin rivativ a) Fin if 5 4 5 36 5 4 5 b) Fin th slop of th tangnt to th curv 36 at th point (, ). Eampl 4: Using Implicit iffrntiation to trmin rivativ g Fin g ' or if 3 a g a g 5 a
Gra (MCV4UE) AP Calculus Pag 3 of 3 Implicit Diffrntiation Eampl 5: Implicit iffrntiations with trigonomtric functions If tan, fin th rivativ of with rspct to. Eampl 6: Highr-Orr rivativ 3 3 If 5, fin ''. Homwork: P. 6 #-4 Cal & Vctors (Optional) P. 564 #, 3c, 4-0,,3
Gra (MCV4UE) AP Calculus Pag of 4 Drivativs of Invrs Trigonomtric Functions Drivativ of th Invrs Trigonomtric Functions (Arc-Trigonomtric) sin cos tan Proof: Arc-sin sin sin ; sin cos If, cos sin sin Proof: Arc-cosin cos cos,0 cos sin If 0 or, cos cos sin Proof: Arc-tangnt tan tan ; tan sc sc tan tan Rcall sin cos cos sin Rcall sin cos sin 0 cos Rcall sc tan tan sc Rcall csc cot cot csc Eampl : Drivativs of Invrs Trigonomtric functions Fin th rivativ of th following sin b) tan a)
Gra (MCV4UE) AP Calculus Pag of 4 Drivativs of Invrs Trigonomtric Functions Eampl : Drivativs of combin Invrs-Trig Functions cos sin an us th rsult to sktch th graph of th rivativs. Diffrntiat Eampl 3: Drivativs of Invrs Trigonomtric functions Fin th rivativ of tan Eampl 4: Applications of Invrs Trigonomtric Functions A particl movs along th -ais so that its position at an tim 0 th particl whn t =6? Distanc is in m, tim in sc. t is t tan t. What is th vlocit of
Gra (MCV4UE) AP Calculus Pag 3 of 4 Drivativs of Invrs Trigonomtric Functions Drivativs of th Othr Thr Invrs Trig Functions sc,, csc,, cot Proof: Arc-scant sc sc ; 0 & sc sc tan sc tan sc an tan sc sc tan sc if if Proof: Arc-coscant csc csc ; 0 & 0 csc csc cot csc cot csc an cot csc csc cot csc if if Proof: Arc-cotangnt cot cot ; 0 cot csc csc csc cot cot cot Eampl 5: Drivativ of th Co-Invrs Trig Functions 4 sc 5 Fin th rivativ of
Gra (MCV4UE) AP Calculus Pag 4 of 4 Drivativs of Invrs Trigonomtric Functions Invrs Function-Invrs Cofunction Intits (Proofs of th intitis can b trmin graphicall) cos sin cot tan csc sc sc cos csc sin Rcall: (a) (b) (c) () () (f) Eampl 6: Tangnt lin to th Invrs cotangnt Curv Fin an quation for th lin tangnt to th graph of cot at. Homwork P. 70 #-9
Gra (MCV4UE) AP Calculus Pag of 6 Drivativs of Eponntial & Logarithmic Functions Dfinition of (Natural Eponntial Numbr) 0 lim(. 78888459... 5 5 5 Proprtis of an ln Rcall th logarithmic function is th invrs of th ponntial function. log b which is thinvrs of b, b log which is thinvrs of. Th function log can b writtn as ln an is call th natural logarithm function. ln Domain is R Rang is R 0 -intrcpt at ln, 0 Horizontal Asmptot 0. log b ln Domain is R 0 Rang is R -intrcpt at ln, R Vrtical Asmptot 0. b = ln Drivativ of f( = Rul (): If, thn. Rul (): If f (, thn f '( g'( Eampl : Drivativ of Eponntial functions Diffrntiat 4 53 3 7 5 a) b) c) 3t ht () 3t Eampl : Conncting th rivativ of an ponntial function to slop of tangnt. 4 Fin all points at which th tangnt to th curv fin b 3 is horizontal.
Gra (MCV4UE) AP Calculus Pag of 6 Drivativs of Eponntial & Logarithmic Functions Dfinition of Natural Logarithm Natural logarithm is th logarithm to th bas. log ln Basic Proprtis ) log 0 ln 0 ) log = ln 3) log ln log 4) ln Drivativ of = ln If = ln, thn. u If = ln u, thn. u Laws of Logarithm ) log log log ln ln ln (Prouct Law) ) log log log ln ln ln (Quotint Law) p 3) log plog ln p pln (Powr Law) Proof : ln ln Eampl 3: Drivativs of Natural Logarithmic functions Diffrntiat a) ln(5 3) b) 4 ln(5 3) c) ln(5 3) 4 Eampl 4: Drivativs of Natural Logarithmic functions Diffrntiat u ln a) h( u) ln u b)
Gra (MCV4UE) AP Calculus Pag 3 of 6 Drivativs of Eponntial & Logarithmic Functions Eampl 5: Drivativs of Natural Logarithmic functions with Laws Diffrntiat a) f ln b) f ( ln (3 )( 5) Eampl 6: Equation of tangnt to th Natural logarithmic function ln at th point whr 0. Fin th quation of th tangnt to th curv fin b Eampl 7: Equation of tangnt to th Natural logarithmic function Fin th quation of th tangnt to th curv fin b that is prpnicular to th lin fin b 4.
Gra (MCV4UE) AP Calculus Pag 4 of 6 Drivativs of Eponntial & Logarithmic Functions Drivativ of Gnral Eponntial functions g ( b g' ln b g0 g ( ln ln b ln gln b ln b g' g b ln b g' Drivativ of Gnral Eponntial functions For g b g ' b lnb g' Eampl 8: Drivativ of gnral ponntial functions Diffrntiat a) 3 5 b) 3 5 4 c) 4ln 4 4 ln5 4 ) f ln8 3
Gra (MCV4UE) AP Calculus Pag 5 of 6 Drivativs of Eponntial & Logarithmic Functions Eampl 9: Solving a problm involving an ponntial mol A biologist is stuing th incras in th population of a particular insct in a provincial park. Th population tripls vr wk. Assum th population continus to incras at this rat. Initiall thr ar 00 inscts. a) Writ an quation to rprsnt th numbr of inscts in t wks. b) Dtrmin th numbr of inscts prsnt aftr 4 wks. c) How fast is th numbr of inscts incrasing i) whn th ar initiall iscovr? ii) at th n of 4 wks? Rcall: Chang of bas log b ln b log a b log a ln a Drivativ of gnral logarithmic function g' For loga g ' g lna Basic Proprtis ) log b 0 ) log b b = 3) log b b log b 4) b Laws of Logarithm ) log log log b ) logb logb logb 3) log plog b p b b b Proof log ' g g a g ' ln a ln g ln a log g log a 0 ln g ln a g' g ln a g g' ln a Eampl 0: Drivativ of gnral logarithmic functions Diffrntiat log 6 4 7 b) log (4 3 ) a) 5
Gra (MCV4UE) AP Calculus Pag 6 of 6 Drivativs of Eponntial & Logarithmic Functions log5 ln ) log 3 5 3 log5 5 c) Homwork: P. 78 #-4, 5, 53 Cal & Vc (Optional) P. 3 # 3, 5 P. 575 #3,4,5ab,6,7a,8, 9a,0-3 P. 40 # 6, 7ab, 8 P. 578 #,, 3a, 4acf, 5, 7, 9a
Gra (MCV4U) Calculus & Vctors Pag of Logarithmic Diffrntiation Logarithmic Diffrntiation Th rivativs of most functions involving ponntial an logarithmic prssions can b trmin b using th mthos that w hav vlop. A function poss nw problms, howvr. Th powr rul cannot b us bcaus th ponnt is not a constant. Th mtho of trmining th rivativ of an ponntial function also cannot b us bcaus th bas is not a constant. In such cass w ar using th logarithmic iffrntiation. Stps in Logarithmic Diffrntiation ) Tak logarithms of both sis of an quation f. ) Diffrntiat implicitl with rspct to. 3) Solv th rsulting quation for. Eampl : Dtrmin th rivativ of a function using logarithmic iffrntiation Dtrmin for th function, 0. Eampl : Dtrmin th rivativ of a function using logarithmic iffrntiation For 3,.
Gra (MCV4U) Calculus & Vctors Pag of Logarithmic Diffrntiation Eampl 3: Dtrmin th rivativ of a function using logarithmic iffrntiation 4 Givn trmin at = -., Homwork: P. 58 # -9,, 3