211-1 Chain Rule

211-1 Chain Rule - Math 211 ws 1 Tutorial Program Chain...

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Unformatted text preview: Math 211 ws 1 Tutorial Program Chain Rule Examples: . . . . 3 7 d9 3 6 2 Goal: To differentlate c0mpos1te functions. 2! = W + 2) g = 7(33 + 2) ' (3m ) . . d 1 Comp051te Functlons: y = f(g(x)) or y = y : «5 i 23: i : E(5 i 2x)—% . (_2) Examples: y = (m3 + 2)7 9(90) 2 1:3 :— 2 y : (3 _ $)—2 fl : _2(3 _ x)—3 _ (_1) = w) dac y 2 m g 2 5 _ 293 Each of these could be simplified after you write the derivative. f 2 fl Combinations: You will often have combinations of product, quotient 1 _2 and chain rules. y I W = (3 ‘ l“) e What is the “inside” function? Example 5 y = $3 1 + $4 gm) 2 3 _ a: for the product rule = 3:3 g(:r) = \/1 + 9:4 Wha»D is the woutsidew function? so f’(:r) : 3x2 but g7 is more complicated. 3 “90217—2 ac: 14—51% s0 llel+x4_Tl-4x3=i d g< > < > g < ) 2< > W Chain Rule for derivatives: = f'(g(:c)) og'(x) 3 a: (i) Differentiate the “outside” functions first with the “inside” y’ = 3332‘ /1 + I4 __ fill function as the variable. V 1 + m (ii) Times the derivative of the “inside” function. Think of peeling off layers by differentiating until you get to the variable, 2 1/1 + x4 from the “outside” to the “inside.” Problems: Differentiate the following. Answers: 3 1) 3(2x + 29% 1) (2x + 3)E 33:2 3 2) —‘ 2) 2r +1 2 933+ 1 —8z3 1 3) — 3) <z4 + 3>3 —2$ ,3/ 4 4) 1—$2 )3(1_$2)§ 5) [t2 + (1 + t)4]5 5[t2 + (1 -- t)4]4(2t + + t)3) 2\/E+ 1 6) ac —— fl 6) — 4\/E an —— fl 7) (3x —— 1)20 -3c 7) (6395 + 1x395 + 1)19 8) [(2132 + 1)3 + 5]4 8) 48$(2l’2 -- U219»?2 + U3 + 513 1 3 1 2 1 _ _ _ _ 1 _ was I) 9>3<x In +332) 2$ — 1 —24x + 23 1 — 10 — 0) (3x+4)5 ) (3x+4)6 11) Find the equation for the line tangent to y = ( :1: 1)2 at (—2, 4) 11) y : 4x + 12 ac l 12) Find the points on the curve y : m2: 1 where the tangent is parallel 12) (1, and (—1,—%) t0 the X—aXiS. ...
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This note was uploaded on 03/27/2008 for the course CS 367 taught by Professor Marvinsolomon during the Spring '08 term at Wisconsin.

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211-1 Chain Rule - Math 211 ws 1 Tutorial Program Chain...

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