DERIVATIVES OF | EXPONENTIAL & ' TRIGONOMETRIC FUNCTIONSmcvau Name: 5.1 Derivati Expon ions, y: ~>) RECALL: Exponential Functions: forction func}on ‘The exponential function isdefinedas: y=" > whereb>Obe1 cee The x-axis (y = 0) is a horizontal asymptote. or' Poet Asspecial kind of exponential function uses “e” as the base, often referred to as Euler's number. This number shows up various times in mathematical calculations. The number ¢ is defined by: j=2_ 2+ 4/423 lim { at e Just as exponential functions have logarithmic functions as their inverse, the function y=e" has y=Inx as its inverse. ‘This function is called the natural logarithm function. Inverse CF g-c™ iD ye Ink y seeeeThe det ive of the exponential function f|x|=e" is flvi=e"_ gi=eh) GD a fixie! f'(x) Using the chain rule: 4=¢ * Example: Differentiate: @HOMEFUN: Pg. 232 # 2ef, 3df, 4c, 7, 11¢, 13Name: = .2 The Derivati he ral mnential Function, y=b" ‘The derivative of an exponential function f|x|=b" is f'\x|=In\b)b" Example 1: Differentiate. a) fix|=3* b) fixi=x?2" Using the chain rule for f|x|=b"™": fixi=g'(x)b?* nb Example a) fi Differentiate. b) gix=8"" Example 3: Find the equation of the tangent line to the graph of f/x)/=2x"(2") at © HOMEFUN: Pg. 240 #1- 3,5, 7abMcvau Name: er: i ion, y=b* ~The derivative of an exponential function f|x|=b* Gin=nioe> Example 1: _ Differentiate. a) fix|=3" b) fixi=x?2" ©) gixi=(4*+x*? $00) = Inale) P10) 22"4 X* Ine(2a)_-gD=% Crex< (aac eax?) Using the chain rule for f| x|= 6?" aan (raceiawne) ce POd= Coo gos Example 2; Differentiate. a) fixina" b) gix|=8""" Pry ny C47*"’) (2) go= tne godoaied = o(q>") (ind) = 20-8 ng, Example 3: Find the equation of the tangent line to the graph of f|x|=2x°(2") at x=1. _ eca> 2003 2") - aay 22 (224 #0) = Gx? (2x7) +2x2(19 (2) (sx?) = Gna(248) 6x5 (In2) (23) cr = Gxt) [+17 (in2)] CD Let (indy) pY.= Om (x-X) Y-H™ 2p. 32 K-22 , G= 26,42 xy -{6%.—_O {Ke ea oF He aeangrere VE - © HOMEFUN: Pg, 240 # 1- 3,5, 7abcvau 4 o* Name: 5.3 Optimization Problems Involving Exponential Functions RECALL: The steps for solving an optimization problem from Unit 3: 1, Read the question carefully and draw a diagram, if appropriate. 2. Determine the equation which is to be maximized or minimized. Substitute into the equation to ensure that there is only 1 variable, then take the derivative. 3. Find the max/min value by setting the derivative equal to zero. 4. Write a concluding statement. Example 1: ‘number, N, of bacteria in a culture at ime, t is $000[30.r62 Find the largest number of bacteria inthe culture during the interval 0<¢<50, where tis measured in hours. N\ N(D= 000+ 1000be N'()* Zecee Ys + 20004 7% Fr G@) G+ 2woee = -leote @=\o0e *B (e-t) o= (10-4) £226 Noe T4, 715 —— 1 He rex H OP Wnctei rer NCQ) = QD Seo ine eottore WM beta NEG= 6B, 1% © HOMEFUN: Pg. 245 # 4a,5,6Mcvau Name: 5.4 The Derivatives of y=sinx and _y=cosx Derivative rules for sine and cosine functions: flxissinx > f\x}=cosx flx\=cosx > f|x)=-sinx g|x\=sinf (x) > glx)=é g\x\=cosf(x)g'[xl=¢ Example 1: Differentiate. a) flx/=x°sinx b) fixl=e“cosx Pi) 2x? (@24) 132 Sax. PIL) =-€ “Bin x + coon (“De ™ =At(KeaeX + SSI») =-e*( singe eo>5) ©) gxissinx® 4) f[x/=cosle") ~ ginzws 0) FD = Caine (€**) )(e* (29) = 2x woo ~? F>2x€** (sine*') °) nle=cs(] 9 fide n= Coo (x4) pC) = CXSiN x — Crcenx Ole tee Gee) De h (= 255 =) =_€* (one ws) : sn 8) fixi=sin(e*) h) fix}=cosé #1L9- > exe") $00) kn (510 X)] Cos x =e*eppe* i) flx)=cos*x F(X)=(Co> x) 4 $1) 24 (eo)? Csinx) = -4 omar Cod 2X © HOMEFUN: Pg. 256 # 1, 2, 3ce, 4b, 5, 6, 11, 12cvau Name: 5.5 The Derivative of y=tanx Derivative Rule for Tangent functions: fix|=tanx > f'|x\=sec’x f[x\=tan g(x) > fix\=sec*g(x)* g'(x) Example 1: Differentiate. a) fixi=x’tanx b) flx)=tanyx741 Fn) xt kann +x? Sce*x er(xy- vec Ke 1 A 1) He x) 0) 2 Cece) Nxt et Example 2: Determine the equation for the tangent line of the function y=tan -|@ = 90S number /natoral Ly 22.718 LE eOd=v*, Kne PU) b* 1b We PA)=0" *Inb+ g@) lim b&-{ = 1nb hao n loner diPProntiakiog @ function hor \Oselve> aN ExpMEMal FLoction, Use (ued, Sunn, ChPR, product A Crain (WED Wren necese tu opkontirg= acrerMINiNg, Varues _ lop jndependenr vewiaie , So Hot the wowe> ¢f te. FUNTION MEAeD We sstuanon (EN LG MO CK MaxIITEA DERIVATIVES - EXPONENTIAL _* NC FUNCTIONS | # Jednnique? Weed to. epromnmee aN Exponenticd Fanenon mockel ace phe same Ob Hee used fo Opremice PAYPOMI AL faxonat functions. - 1 underatand — auanie>, — O varieine For Gvannry. 2, delermine domain 3, algonWamn Foc ExONE VOWED “> MO 4 MN. GN dst 4, Step D Veoulk For anower. 5. canes SSDS * A029 - Gos x @ AIGA. -s0 ax ax + yz ain 9H), then BF = aps MP + y= 00 600, ton BE = an FLVEO “laren ciPPreNAKING G FeNction wai Ponctione , vse Chain es gam, CNA, Quotrent when necledt = ddan X) . sect x ax © * Fe Gran POD) = sec* POD* FOO, “Hig dentties Can be Weck to corte 4 EXpteoaon ce an @piuilenr Exp drPrenvate. somonine? new Foo a (camer to wore1 pe seem 2 an 7 5 2 (xa) ak + 2 De WS 4. yo pkang 2-9 Be -a9ee%"-~-X) A (7%) ee / SCae Hn qth =4Lnd (3)4 = unxt ae eee Po dg. syrya 2 2x-ln 2 ox ey Be 3- Bink? Se ~ \=2 nx Bet 2 rt inxt But =te2x eoo e* G gnjre™-% D (nj) = e% £*%2 OD no nin 7 4 Sx (on A a dx Corx TEsN @xew -m =O ay M EL 2 ron B. Fly\~ = ~ x ) | x12%eIne- 2% PW oe ROS. ling- 2% =® KLIN) ew KinZ=4 X= Far Ble Ye tan * c dy > 2 tan(e”) se (ASE = rehtan(e%) (2%) 10. Pln)=e2* 9 D erm) ae Le ™* rteMCV4U UNIT 5: DERIVATIVES OF EXPONENTIAL AND. TRIGONOMETRIC FUNCTIONS DAY TOPIC SECTION HOMEWORK | Derivatives of Exponential 51 | Pa 282 # 2ef, 3df, 4c, 7, Ie, Functions, y=e" 13 The Derivative of the 2. | General Exponential 5.2 | Pg. 240#1-3,5, 7ab Function, y=b* Optimization Problems 3. | Involving Exponential 53 | Pg, 245 #40,5,6 Functions The Derivatives of y=sinx Pg, 256 # 1, 2, 3ce, 4b, 5, 6, 11, 4 }e 54 | * | and y=cosx 12 5 | The Derivative of y=tanx 55 fe Be coe Oe oe | 6 Pg, 263 - 265 # 1-23 REVIEW Pg, 266 #1-9 7 8 | UNIT TEST #5