Chapter 3 Section 08

Chapter 3 Section 08 - - xsin(xy)] ) 2ysin(y - [xcos(y) y...

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1 3.8 Implicit Differentiation page 180

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2 Objectives 1. Find the derivative of implicitly defined equations. 2. Find the equation of a line tangent to an implicitly defined curve.
3 Examples -4 -3 -2 -1 1 2 3 4 5 -4 -3 -2 -1 1 2 3 4 x y x = sin(y) -1 1 -1 1 x y x+y=(xx+yy)^2 Some functions cannot be defined explicitly as a function of x. That is, we cannot get “y” on one side and a function of just “x” on the other side. These functions are defined implicitly by an equation relating x and y.

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4 Implicit Differentiation We do not have to solve for y to find the derivative. Implicit differentiation consists of differentiating both sides of the equation with respect to x and solving the resulting equation for y’.
5 [ ] [ ] [ ] [ ] [ ] ' ' 2 1 ' ) cos( ) sin( ' 1 1 ' 2 ' 2 2 y e e dx d y y y dx d y y y dx d y y y dx d y y y dx d y y dx d y y = = = - = = =

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6 y x y y x y x yy yy x y x Example 9 4 ' 18 8 ' 8 ' 18 0 ' 18 8 36 9 4 : 2 2 - = - = - = = + = +
7 x y x y y x y x y y x y xy y y y y y xy x y y y xy x y xy x Example 2 3 2 2 ' 2 2 ) 2 3 ( ' 2 2 ' 2 ' 3 0 ' 3 2 ' 2 2 0 ' 3 )] 2 ( ' 2 [ 2 1 2 : 2 2 2 2 2 3 2 - - = - = - - = - = + - - = + + - = + -

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8 xsin(xy)     ) 2ysin(y   -   xcos(y) sin(y)   -   ysin(xy) -     y’ sin(y)   -   ysin(xy)

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Unformatted text preview: - xsin(xy)] ) 2ysin(y - [xcos(y) y sin(y) - ysin(xy)- xsin(xy)y )y 2ysin(y - xcos(y)y ysin(xy) - xsin(xy)y- )y 2ysin(y - sin(y) xcos(y)y y(1)] sin(xy)[xy- ] )2yy [-sin(y sin(y)(1)] [xcos(y)y cos(xy) ) cos(y xsin(y) ation. differenti implicit using derivative the Find 2 2 2 2 2 2 + = = + = + = + + = + + = + 9 Tangent line ) 1 ( 2 1 2 4 2 1 2 4 ) 1 ( 4 | ' 4 2 36 9 2 ' 9 2 36 ' 2 36 ' 2 9 2 : slope find ) 1 )( 1 ( ' 2 4-y ). 2 4 (1, at 1 36 y 9 x o tangent t line the of equation the Find ) 2 4 , 1 ( 2 2--=--=-=-= -=-= = +-= = + x y y y x y x y x yy yy x x f 10- 8- 7- 6- 5- 4- 3- 2- 1 1 2 3 4 5 6 7 8- 8- 7- 6- 5- 4- 3- 2- 1 1 2 3 4 5 6 7...
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This note was uploaded on 09/28/2009 for the course MATH 1550 taught by Professor Wei during the Spring '08 term at LSU.

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Chapter 3 Section 08 - - xsin(xy)] ) 2ysin(y - [xcos(y) y...

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