Explanation Given information: The provided equation is y ln x + y = 2 . Formula used: 1) Product rule of derivative of differentiable functions, f ( x ) and g ( x ) is d d x [ f ( x ) ⋅ g ( x ) ] = f ′ ( x ) ⋅ g ( x ) + f ( x ) ⋅ g ′ ( x ) . 2) The derivative of function f ( x ) = u n using the chain rule is f ′ ( x ) = d d x ( u n ) = n u n − 1 d u d x , Where, u is the function of x . 3) The derivative of natural logarithm of x is d d x ln x = 1 x . 4) Constant multiple rule of derivative of function f ( x ) is f ' ( c x ) = c f ' ( x ) Where, c is constant. Calculation: Consider the equation, y ln x + y = 2 Take d d x of both sides, d d x ( y ln x + y ) = d d x ( 2 ) d d x ( y ln x ) + d y d x = d d x ( 2 ) Apply the product rule of derivative and simplify the derivative, d y d x ⋅ ln x + y ⋅ d d x ln x + d y d x = 0 ln x d y d x + y ⋅ 1 x + d y d x = 0 ln x d y d x + y x + d y d x = 0 Evaluate the value of d y d x by isolating it to the left side, ln x d y d x + d y d x = 0 − y x = − y x Take out d y d x as common, ( ln x + 1 ) d y d x = − y x Simplify the derivative through dividing both sides by ( ln x + 1 ) , ( ln x + 1 ) ( ln x + 1 ) d y d x = − y x ( ln x + 1 ) d y d x = − y x ( ln x + 1 ) Thus, the value of d y d x for equation y ln x + y = 2 is − y x ( ln x + 1 ) using the implicit differentiation

Student Solutions Manual for Waner/Costenoble's Finite Math and Applied Calculus, 7th

7th Edition

ISBN: 9781337275972

Author: Waner, Stefan; Costenoble, Steven

Publisher: Cengage Learning

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Chapter 11.6, Problem 9E

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To calculate: The value of dydx for the equation ylnx+y=2 using the implicit differentiation. Also compare the result obtained by rearranging the equation for y as a function of x and taking the derivative.

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Chapter 11.6, Problem 8E

Chapter 11.6, Problem 10E

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Chapter 11 Solutions

Student Solutions Manual for Waner/Costenoble's Finite Math and Applied Calculus, 7th

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The value of d y d x for the equation y ln x + y = 2 using the implicit differentiation . Also compare the result obtained by rearranging the equation for y as a function of x and taking the derivative.

Solution Summary: The author calculates the value of -yx(mathrmlny+1) using the implicit differentiation.

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To calculate: The value of dydx for the equation ylnx+y=2 using the implicit differentiation. Also compare the result obtained by rearranging the equation for y as a function of x and taking the derivative.

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Chapter 11 Solutions