The Chain Rule 3.5.Chain Rule Suppose that y is a differentiable function of u and u in turn is a differentiable function of x.Then y is a composite function of x.Does y have a derivative with respect tox?Yes it does,and dy/dx is given by a formula that is easy to remember: (3.5.1) dy dy du dx du dx Leibnitz form of the chain rule,p.133 Main Menu l
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Chain Rule Leibnitz form of the chain rule, p. 133 3.5. Chain Rule
The Chain Rule Example 1 Find dy/dx by the chain rule given that y=41 u+2 and 1u=x2 Example 1,p.133 Main Menu c5时
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Chain Rule Example 1, p. 133 Example 1 Find dy/dx by the chain rule given that y = and u = x 2 . u – 1 u + 2
The Chain Rule Example 2 Example 2,p.135 Main Meny 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Chain Rule Example 2, p. 135 Example 2 + −3 1 x x dx d
The Chain Rule Example 3 d [1+(2+3x)5]3 dx Example 3,p.135 Main Menu 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Chain Rule Example 3, p. 135 Example 3 d dx [ 1 + (2 + 3x) 5 ] 3
The Chain Rule Example 4 Calculate the derivative of f(x)=2x3(x2-3)4. Example 4,p.135 Main Menu c8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Chain Rule Example 4, p. 135 Example 4 Calculate the derivative of f ( x ) = 2 x 3 ( x 2 – 3 ) 4
The Chain Rule Example 5 Find dy given that y=3u+1.u=x=1-s. ds Example 5,p.135 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Chain Rule Example 5, p. 135 Example 5 Find given that y = 3u + 1, u = x –2 , x = 1 – s. ds dy
The Chain Rule Example 6 Find 少 at t=9 given that dt u+2 J=_ =(3s-7y,s=V7 Example 6,p.136 Main Menu 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Chain Rule Example 6, p. 136 Example 6 Find at t = 9 given that y = , u = (3s – 7)2 , s = √t. u + 2 u – 1 dt dy
The Chain Rule Example 7 Gravel is being poured by a conveyor onto a conical pile at the constant rate of 60 cubic feet per minute.Frictional forces within the pile are such that the height is always two-thirds of the radius.How fast is the radius of the pile changing at the instant the radius is 5 feet? Example 7,p.136 Main Menu C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Chain Rule Example 7, p. 136 Example 7 Gravel is being poured by a conveyor onto a conical pile at the constant rate of 60π cubic feet per minute. Frictional forces within the pile are such that the height is always two-thirds of the radius. How fast is the radius of the pile changing at the instant the radius is 5 feet?
The Chain Rule THEOREM 3.5.6 THE CHAIN-RULE THEOREM Ifg is differentiable atx and f is differentiable at g(x),then the composition fog is differentiable atx and (fog)(x)=f(g(x))g(x). The chain-rule theorem (3.5.6),p.138 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Chain Rule The chain-rule theorem (3.5.6), p. 138