Implicit Differentiation;Rational Powers Example 1 We know that the function y=1-x satisfies the equation x2+y2=1. y=1-x2 (-1.0) (1.0】 2+2=1 Example 1,p.147,Figures 3.71-2 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Implicit Differentiation; Rational Powers Example 1, p. 147, Figures 3.71–2
Implicit Differentiation:Rational Powers Example 2 Assume that y is a differentiable function of x which satisfies the given equation. Use implicit differentiation to expressd in terms ofx andy. (a)2xy-y3+1=x+2y.(b)cos (x-y)=(2x+1)3y. Example 2,p.147 Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Implicit Differentiation; Rational Powers Example 2, p. 147 Example 2 Assume that y is a differentiable function of x which satisfies the given equation. Use implicit differentiation to express dy/dx in terms of x and y. (a) 2x2y – y 3 + 1 = x + 2y, (b) cos (x – y) = (2x + 1)3y
Implicit Differentiation:Rational Powers Example 3 Figure 3.7.3 show the curve 2x3+2y3=9xy and the tangent line at the point(1,2). What is the slope of the tangent line at that point? 1,2 -3-2-1 12 -2 2x3+2y3=9y Figure 3.7.3 Example 3,p.147-148,Figures 3.7.3 Main Menu C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Implicit Differentiation; Rational Powers Example 3, p. 147-148, Figures 3.7.3 Example 3 Figure 3.7.3 show the curve 2x3 + 2y3 = 9xy and the tangent line at the point (1, 2). What is the slope of the tangent line at that point?
Implicit Differentiation:Rational Powers Example 4 The functiony=(4+x3 satisfies differentiation y3-x2=4 Use implicit differentiation to express y/dx2 in terms ofx andy. Example 4,p.148 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Implicit Differentiation; Rational Powers Example 4, p. 148 Example 4 The function y = (4 + x2 ) 1/3 satisfies differentiation y 3 – x 2 = 4. Use implicit differentiation to express d 2y/dx2 in terms of x and y
Implicit Differentiation:Rational Powers Ifn is a positive integer,then we have d(）=x The formula can then be extended to all rational exponents p/q: d (3.7.10 (xP/)Px(P/q)-1 dx The derivative ofrational powers,(3.7.1),p.149 Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Implicit Differentiation; Rational Powers The derivative of rational powers, (3.7.1), p. 149 If n is a positive integer, then we have ( ) . −1 = n n x nx dx d
Implicit Differentiation:Rational Powers If u is a differentiable function ofx,then,by the chain rule d (3.7.2) 卫pg-1d Chain-rule version,(3.7.2),p.150 Main Menu C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Implicit Differentiation; Rational Powers Chain-rule version, (3.7.2), p. 150
Implicit Differentiation;Rational Powers Example 5 d (a),[(1+x2)5] dx ⑥)4[0+x2)29] dx d [(1-x2)14] dx Example 5,p.150 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Implicit Differentiation; Rational Powers Example 5, p. 150 Example 5 d dx (a) [ ( 1 + x 2 ) 1/5 ] d dx (b) [ (1 + x 2 ) 2/3 ] d dx (c) [ ( 1 – x 2 ) 1/4 ]
Implicit Differentiation;Rational Powers Example 6 [()] Example 6,p.150 Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Implicit Differentiation; Rational Powers Example 6, p. 150 Example 6 d dx [( ) 1/2 ] x 1 + x 2