Explanation Given: The equation of the curve is y 3 = 8 x 4 . Also, the point is ( 1 , 2 ) . Formula used: Chain Rule: If f and g are functions and y = [ g ( x ) ] , then y ′ = f ′ [ g ( x ) ] ⋅ g ′ ( x ) provided that f ′ [ g ( x ) ] and g ′ ( x ) exists. And, The equation of tangent line is given as: y − y 1 = m ( x − x 1 ) Here, m is the slope, x 1 and y 1 are the given points. The equation for slope is: m = d y d x . Calculation: Implicit Differentiation: This differentiation is used to evaluate d y d x , when y is defined implicitly as a function of x . The differentiation is carried out in two steps: Step 1: Both the sides of the given equation are differentiated with respect to x . Since y is the function of x , always use the chain rule. d d x ( y 3 ) = d d x ( 8 x 4 ) 3 y 2 d y d x = 8 d d x ( x 4 ) 3 y 2 d y d x = 8 ( 4 x 3 ) 3 y 2 d y d x = 32 x 3 Step 2: Solve the resultant equation for d y d x

Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)

12th Edition

ISBN: 9780134767628

Author: Margaret L. Lial, Thomas W. Hungerford, John P. Holcomb, Bernadette Mullins

Publisher: PEARSON

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Chapter 12.4, Problem 31E

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To calculate: The equation of the tangent line at the given point (1,2) for the given curve, y3=8x4.

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Chapter 12.4, Problem 30E

Chapter 12.4, Problem 32E

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Chapter 12 Solutions

Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)

Ch. 12.1 - Checkpoint 1 For what values of x is the function...Ch. 12.1 - Checkpoint 2 Find all intervals on which is...Ch. 12.1 - Checkpoint 3 Identity the x-values of all points...Ch. 12.1 - Checkpoint 4 Find the critical numbers for each of...Ch. 12.1 - Prob. 5CP Ch. 12.1 - Prob. 6CP Ch. 12.1 - Checkpoint 7 Find the locations of the local...Ch. 12.1 - Prob. 8CP Ch. 12.1 - Checkpoint 9 If a sales function is given by...Ch. 12.1 - Prob. 1E

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The equation of the tangent line at the given point ( 1 , 2 ) for the given curve, y 3 = 8 x 4 .

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To calculate: The equation of the tangent line at the given point (1,2) for the given curve, y3=8x4.

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Chapter 12 Solutions