University:
Felician University
Course:
MATH-002 | Basic Math Skills-Algebra
Academic year:

2020
Views:

284

Pages:

4
Author:

SpaceCadet101

ТНЕ CHAIN RULE NOТES ASSIGNMENTS: CHAIN RULE WEB ASSIGN AND PINK CHAIN .RULE WORKSHEEТ Objectives: ■ ■ ■ Find the derivative of а composite function using the Chain Rule. Find the derivative of а function using the General Power Rule. Find the derivative of а trigonometric function using the Chain Rule. The Chain Rule THEOREM 2.11 The Chain Rule If у = f(и) is а ditferentiaЫe function of и and u = g(x) is а differentiaЫe tunction of х, then у = f(g(.x)) is а ditfeгentiaЫe fш1ctioп of .х and dy _ dy d.x du du dx ---•- or, equiva1ently, cl dx[J(g(x))] = f'(g(x))g'(x). you_Q.... т LD\AЛ tc. tJvr� 0� В -------- When applying the Chain Rule, it is helpful to think of the composite function f g as having two parts-an inner part O and an outer рагt. The derivative of у= f (и) is the derivative of the outer function (at the inner function и) times the derivative of the inner function. OL1tc1· l"t111c1it)Jl У / \ \ / = J(!.{(x)) = /(11) у' = .Г(и) • и' The function у= (х2 + 1) 3 is an example of one of the most common types of composite functions, у= [u(x)] n . The rute for differentiating such functions is called the General Power Rule, and it is а special case of the Chain Rule. zx THEOREM 2.12 The General Power Rule If у = [и(х)]n, \vhere u is а cliПerentiaЫe f'пnction of х and п is а гatioпal nшnber, then с/у clu = n[u(x)Jn- 1 dx clx -=- or, eqнivalently, Examples Set 1: Find the derivative of each. + Зх 3 - 18х 2 + 32) 8 а. /(х)={Зх-2х2) 3 Ь. у= 10(4х 4 с. h(x) = ✓1 - х 2 ::: d. f(x) = 1/-бх 3 -х ______ \_ t-)t 2.. +7 - Trigonometric Functions and the Chain Rule The "Chain Rule versions" of the derivatives of the six trigonometric functions аге as follows. d du 2 du [tan и] = (sec и)и' .!!.._ [cos и] = -(sin u)u' du :)sin и] = (cos и)и' [cot и] = -(csc2 u)u' d du d [sec и] = (sec и tan u)u' Example Set 2: Find the derivative of each. d du [csc и] = -(csc и cot u)u' Ь. у= tan (Зх 2 - 1) а. у = sin 2х - StY) '-:ZX l � Examples Set 3 Repeated Application of the Chain Rule: Find the derivative of each. а. h(x) = х 2 ✓1 - х 2 = Х • 2 �C,,-r l�UD� et.Jt-€ �AI 6 -� ½. 1Ь (Зх-1) 2 - (х2+з)2 • у J QvOT\Г �L€, \� •. С-\д� � [�L\-\.)t) �.. т· : � _ cl +i)(z.i.,3'x.�(з - эx-1Ji�6l:.� 'Z, ' 1 - ��-+3) 11' ,: �- (:j--e, �--+3J lf \ \ ( \ :2.. 2 "2. - Чхf._Э(.---t [У. :i-3 • с. у = sin 3 (4t) - � 1 n(4t� 'L '3 d. у = csc 2 (7x) - Неге is а summary of all the differentiation rules we have learned to far: Summary of Differentiation Rules General Differe11tiatio11 R11les Let и and 11 Ье dШerentiaЫe fпnctioпs of х. -с/ [с] = О, с 1s а геаl пвтЬег. GX1 (Simple)Po1�·eг Rule: rl . ti -[х"] = 11.х" -1, -[х] = l, 11 is arational nвniЬer. dx rlx Consшnt M11/tipl