Calculus I - Module 5 Differentiation

formulas Questions Module Differentiation 5 Theslope of the tangentline Frompointslope form man y f Solving Ilya flyfacalIh's is at point My a a fear x a Example 1 letFox X't I b Considerthepoint pal 2 onthe Curve 1 findtheSlope 2 findthe equation ofthe tangent line man Ima flyfatal at Cgifeat I Hythe Man 72 I toformso ca 1 FctortheoremFoyenpoiyg.gs fcxsxs ifcy.in Sythetic Example 2 PETTIES SlopeOF g Stoagenaffne given n'Mo Y o pgppcagg.f Fasa KI I I s divisions comes try 1115Iast The derivative of a function F at X is given by f x link Fate thefunction is differentiable If the limit existwesay theprocessoffindingthederivativeof afunctioniscalled differentiationthederivative fix isread as f primeof X Othernotions Of Y FCx F X y f GI Example I derivative findthe fix link FA 42 3 Of Fox FG 40 him II 4 limo n'To fed goaltofind h in numerator 36 2 3 toform E4x23 x't_hp Ht8xy 4h Nyt 4 31 12 Plug in ble den O Example 2 find thederivative of Fox I fix limo FathIFA limo FE 2L E Inf toform I E th I is t s d Ing Example 3 Sketchthe graph of fromthegraph off of I s FIX fatty link so 36th 5 3 X 2 4 425 15 3 f x limnso 6 I 5 FIX 3 501 gygg Slope tangentline at Example 4 FX 3 2 8mh 2 6 4 342 34 6 2 5 I 3 2 5 Ath 4th 5 6 310 4434 46 131 5 2 Basic differentiation Rules b Rates dldx Constant Rule 0 c dldxEx Power Rue Nx Mdx Cfa Cf ConstantMultiple Rue x Linear Combo Rule d dx af x I by x leaving Find the of Change af 1 bg x x derivatives Constant Rule a d dx a b d dx a Max o 527 O b Mdx x C Old e D c dldx x d4 T 100 4 f gt 37 g S t 4T D x g CY b Mdw 4 2 3 C F x Tx dy FF 2 1 4 2 43 8 3 3zef.si bei ddwczew E x Kt 76093 107 9 3E E S E IF 4 34 EF 10x g S E TE SE O Mdx x 1 3 Ffax E O Constant MultipleRule AMdx 2 3 ddldEE 6 0 C FX O F 0 x2 2 3 3 at E EE s E 1 I Linear Combo Rule constant fit daff ddx aFox x 4 2 day bf x 4 3 3 4 77 10 9,133 2 E I 413 31IE d 752 5 E 12 x 743 7 12 2 3 7 3 5 x 4 x 3 3 x 7 x Ee É O ZE 3EEE 2 42 to 7 0 43 4 0 92 8 7 It ECEHxare e i Tx Fx10 3 42 3 3 2 4 2x 7 CgetsIt E 2ft 43 4 113 x thx T 5Mdx x 5 t ZEE E i É Mdx a 0 24 3 13 43 5 5 Finding an Equation of a Tangent Line d find Tofind F 2x Fox 2 2 X S X l x equator Fa 245 78 4 1 2 2 x 5 at Cl 6 tangentlinetothegraphFox slope F point 5 I 4 1 M 443 1 x 3 1 Point Slope form for equation y y y 6 Mix xp 3 x 1 4 3 3 6 3 4 36 b let F x 2 3 2 2461 O 6 2 24 F FG fp X 3 24 3 forwhatvalues of x 2 3 24 6 x 2 3 24 O ble slope of 6 2 24 0 61 2 41 0 x 4 4 0 x2 4 IF horizontal the Iz D 20 2 is thelinetangent tothegraph f horizontal Find Derivative UsingDifferiationrules of transcendental functions Derivativerulesfor trigonometric functions Mdx sinx Cosx Mdx cosx Six Hot es csexcot .is d x sect seetan dldxLexJ e e dx csex 0 forNaturallogarithmicfunctions Derivativerules E Mdxflux Mdx In1 17 X 0 LearntFindthe 1 derivatives Trigonometricfunctions 2 2 3sinx 2cosy te A y e3 2 x2 3 Smx 2 cosx 2 2x 3 Cox 2 Smx 0 4 3cosx 2 six b y 3tanx 2secx Sx 3Mdx Eaux 2 Mdx sex 5 3 seex 3sec x CY St 2 sextant Sx 2 sextant 5 Cott Beset 4 3 Mdx E x Mdx Cott 3 6 Csce Mdx4 3CCSCECote O CSCE 514 3 ZOE CSCE 3CSCECott 0 Natural Exponential functions A F x Six 30 9 Mdx Sinx 3My ex cosx 3 ex O Cosx day b 3s in x 30 Ze t5 3 3 3 Osx 2 et Es 3 3 cos x Zet 5 x 3 Natural logarithmic functions b Y 2ex 3 tnx 25 2 Mdx ex 3 6 tnx 2Mdx e 2 ex 2 E 3 k 2ex 31 Zet 31 CF X 212 0 53 3 3Mdx Six 2Mdx ex 3 Mdx x2 3 OSX 20 5 3 63 3cosx 20 5 2 4 33 3 2 7 3 six zet 3 3 5 3 35 Find the higher order derivatives y F x Yax lay fat D CY Secondderivative y f x Yax d y fx DILY third derivative y F x Ydx3 dldx3 f x p3x y Fourthderivative y F lx dyldx dYdx4 fX nthderivative yin P x d'Ydxs Mdxs fX firstderivative Dx y DSx y leafy Find the 3rd derivative A Fox 5 72 5 524 3 712 3 0 F x 20 3 14x 4 2013 2 1411 F x 60 2 14 60 2x pg b y 3smx 20 7 F x P x F 3 3cosx 25 763 2 3cosx 25 21 3 3sinx 20 21 2x 3sinx 25 42 3Cosx 20 424 x 3Cosx 20 42 Find using derivatives rates of change Position b Velocity E time is givenby thepositionformula s fCt AS Flat At Fla location of a moving location at Displacement Averagevelocity Velocity instant Velocity Speed Aslate flat Ucasglim Fca fcatyy fcas f.ca P Acceleration dhe F Velocity vets Speed Nt If't I Acceleration act t Mdt d HE p e Ball is thrown up with initialvelocity 64ft seefromansoft building WtfAThe Sec position oftheball after t Sct 165 646 80 A Findavg Velocity at time interval 1,23 b FindVelocity P Acceleration functions C Highestpointreached by ball d Timerequiredforball toreach ground CFindVelocity of ball A Ag F Ela b a at impact F 2 F I L16125 642218,03 164 644180 1128 1441 16 ft Sec B V f t A F t 165 64 80 160 664 64 6Ct 80 16 20 644 0 V FC 326 64 32Mt t 64 32 1 to AFE CTo find this set 0 32 Velocity 0 solvefor t 326 64 31 32 2sec t D Tofind thetime Set Sct 0 solve for t 168 641 80 0 16 E 4t 51 0 E 4 5 0 E 5 th tIsec or E To Find Velocity 326 64 3215 64 96ft see O Isee Substitute E5 in velocity function ult 5.3 b Quotient Rule product usingproduct Rule to find Derivative of a function Theproduct Rule 4dx f x gxD Fa g'G dldxCuv Alternate form golf'd or't Vu LeafProduct Rule a S 7 33 6 6 33 6 33 6 5 7 5 7 519 2 6 3 3 6 72 7 45 230 63 3 42 21 3 4 2x 84 3 45 2 84 30 1 Alternate form 0 5 7 U't b Es int 7 V 3 3 6x via 2 6 F Uv X x 5 Tx Vu 5 7 719 2 6 t 3 3 6 367 45 2 63 3 30 42 21 3 42 F x 84 3 45 2 84 30 E Mdtsine SintMdt E E cost Sint Zt Ecostt 2tsint E Ecost 2Sine CGox 3ex 5x thnx U 3et V 5 2 thnx y Ze u's ur two x g 35 lox E x 5 2 tnx 30 3ex lox 4 5 2thnx 3ex 5 2 10 4 thnx fix gaffe g x A fax golf 83 6 3x et 3 FA 3 8 83 6 3 1 2 8 677 3 0 8.3 0 83 6 Gl E B gx 28 1 cost g D y x 28 1 Cox zeta G E O Osx Ze t 1 setSsinx 2tnx y E4 Product Rule x S X 0 s sinx 24 CSisin x Zenx set E Osx Smx t St 41,5 EYJ Set C 3 3 4tanx Y 33 4 9x tanx 33 4 tanxt 3 3 4 tanx Seex Using quotient rule Quotient rule Eff Alternate form 965fqfyyig.es I vigor v58 Quotient Rule A P 34 x 2 12 1 4 1,712 ÉI b In Icsa zÉyz csexcotx cqgyc 2xcsggtyicsexcot xeff.to's C.IE qe YEE Alternate form 0 45 1 V TE Gt U St vine a I É LE islu.ca 94ftgg 5663 725 656814636 942 365 146 9 Eat 563 74 6444636 9 Lt gg ex F'GEEKg A Quotientrule Figgg get EIox Exffxox g XXIII 5 5 2 É B or Eg a PMS P'Cx 10.1 2415.504 xs ÉIosx SMIGIEL cos FI.gg SMX sinx CosXEEx2sinx e fits 2 x YI I 2cosx O 2sin 2 65 1 2 F Ct cents GET lnts 2k Gt Yt Rule power Int 2548811nts 2.35 E 12,449,4112 I Recognize when they can A Y 2 2 2 Y 2 3 2 43 2 2 be simplified 4 21 3 2 X2 2 y 32 X2 2 3 G 2x 33 72 3 LUTETIA Easier Y Y 12 3 24 3 41 correct in form ng putbefogged'd y 22 2 2 32 x2 2 3 3 2x cat EFFIE IS 812 12 3 Y 4 IS Icy 2 41 2 W on 4 2131 6 4 3 O 24 3 3kt 41 2 4 4 6 4 2 3 6 4 2 4 X 2 46 Simplify X 1 d Solve 2 x B hCw I X 2wq9w W t 3 EW I L 2.21 4W Muy 0 t 63 Ws 4 91 why 2W 9 W3 3 W Y 3 4 W'S Formula Sheet Slope FAEElas Limit Definition Equation of line Fat y y Product Rule Fox Quotient Rule go Fax m go x x go fix FIFES ga Avg velocity Flbjala