Module 6 - The Chain Rule

6.1 ChainRule Thederivative of A y flu can be expressed as É at B F gas g'Cx fegex For the composite function y fog x f g Outer function inner function General Power Rule To find derivative if y A4 2 d 1 n f F Cx x 3 Max 2 dx 1312 4 B Yd Fett dx If then fix 1 35 5 4 dldx 3ZE t et 421st'tet C Y S in x y Sind Idx 3 six Mdx Six 3s n'x COSX 1 F x 513 15 2013 155.4 20 3 155 IIs 3 3 dx 1312 dy lax 1712 2 26 2 1 dydy Yz 3 44 1 2 Gt et L D FAS 5 3 d 1 Chain Rule For F V dldx function x Sino 1 1 d dx d dx d dx 8 1 Mdx p d dx Coto d dx esco Lest u u Cscucotulu constant Sin Kx K COS KX k CSC KxCotKX CSC KX Seckx d dx LOS Kx kseckxtankx K Sin RX KSeckx Kise 2K X tanks d dx accosus sinus Osu v 1 If k is Trig Functions cot Kx Find derivative AS y Sin x2 flt tan 3 24 C Y cos Lettanx B A y B Flt y Sin x2 LOS X Yax x2 COS x 2x 2x COS X Ean 3 Zt Sec 3 26 3 28 See 3 2 3 662 GE Sec 3 ZE C y Cos Lettanx CosCe'tEanxiettanx Cosletttanx CetSec E Sec Since'ttant Chain Rule For Exponential t logarithmic d dx eft F era ddx In Foxy or x Fox F Max e e or Mdx x Fa v u luv v u u f 0 x IF U O then dldxlenlul Find derivative b A y II e e y's3,13 C FIX 7 x 2 2 E 3x e ecost Sint Sint 1 Cayee't 2 Int Eisen y cost lost B F E X z eL5 p I fittest ethic 1 2112 7 1 4 1 Ex is E Y Cent 3Cent YE Ent lnlxytnl.mxlnxtlnylnltyl Chex ln lnxh A y Inlxet thnx the Lnxt2x Yx 2 X B yen 1 3 2 thx ln 3 tnx 3 x lnx C y lnFx1 3 2 3 In1 2 3 2 3.2 4 412 1 Y2ln 2 42.1 2 2 1 1 1 Exponential b logarithmic wi bases otherthan e d dx d dx Eb b day Clogbx Mdx o lnb b lnb b V lnb lx lnb Yo logbu v Find derivative A Y x2 B Y 2 Zx Cln2 2 g 2 2 ti ln5 5 L4ln5 x5 C Y log Ux OSX en3 Yuu lenz Yen Lost CSinx cost tanx3 V lnb x lnb u A 1 IF has g x t 1 1 2fCx Find h 4 I C pox Chain Roleplex valuate at 3 f p Ci gas g'd gci g'd 6 Faget F P'c so F 10.6 d MEX P mix n'co COSXS Ficosx Note Regas 60 M Co Cox P'CCOSXIC Sinx ChainRule P'Ositsin P'Ll O fax gas C gexs e G'Cx g 4 fat gica efa pic Efa C C ex EY 3 era 9164 efa f aus e0 D at F y 3 9 4 6.2 Implicit Differentiation To Find derivatives of implicit functions 1 Apply dldx on both sides of the equation 2 Distributedldx to each term 3 Differentiate 4 Allterms w d4 dx on left Others on aright 5 Factor d dx out from left side 6 Solve for d dx 2 find Idx For y 242 34 x2 at 2,1 lax L 2 Idx y 24 3y x2 4 4 Mdx 34 Mdx x2 Mdx 2 4 2 Maxcy 3y2dHdy 4 Ydy 344dg 2x 0 2x 342 44 3 o JETT Slope o find d 32k dx for 43 2 4 43 9 ya x2 3 2 Idx 9 342 x2 Mdx a Mdx Mdx X dldxCy3 dldx2xy 94 1 y 2 3yldYdx 2y 2xdYdx 3 2ydYdx 2x 2 342 2 24 3 2y 2x dx y 2 4 x SEIZE Finding Y o equations Ma of tangent lineusingimplicit Differentiation Y X y Xy t 42 12 2 2 on Curve a Show d b find dx usingimplicit differentiation C findequation of line to curve at Point L2 2 d Findanypoint where line is horizontal C Find any point where line is vertica A 2 2 2 2 4 L4 2 2 12 4 12 4 4 4 12 Mdx B laxly Xy x2 Mdx y'd Mdx Zy C L X iz Xy thx x2 Y 2x I EEE 4 2 Eff 2x 4 2 Y L2 mix y y 0 7 12 O X 2x 2 27 4 2 4 3 2 4 X 2 71 0 2 1 X2 2 m 12 13 14 12 2,471 422 E Zy X O 42 4 y 242 42 X Zy 2472 442 12 12 By 4 14 4,2 PL4 2 6 3 Related rate 1 Writevariables b Sketch 2 Equation 3 Identify rate of Change Differentiate implicity on both sides w respect to all knownvariables into equation Solve for rate of Change 4gSubstitute time Suppose a circle with radius r is increasing by 25msec A whate rate is the area changing when the radius is 4m or Idt 25msec re ret da A A t at r 4m A rt datA Idt 2Tir Tra 2T 4m C 25msec 27m21sec 0 A loft ladder leans against when the top How fast is a wall overtime the ladder slidesdow is 8ft above ground the bottom is slidingawayat 2f the ladder falling k g 2ft see 1 set h't 12 100 105 2h 21 0 12 100 64185 365 WE 21 2h f f f 1.2 15ft see related rates WI 3 deminsion objects A snowball is melting When the volume has Shrunk 367 em's the snow is melting at a rate 0.1cm min at what rate is the radius changing 5 3671 4 27 v3 Clyde v 3 Tr is V V t If When V 367cm 7 F 3cm Mdt 4 3763 dude 4 Tr Mdt Yat 4T 3cm lat III du de 0.0087 COLem cm im in min V Trh 4 3 A Trl hit l Y Tv3 X x2 22