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Chapter 9.6, Problem 24E
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Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Solutions

Chapter
Section
BuyFindarrow_forward

Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

At the indicated point, for each function in Problems 21-24, find(a) the slope of the tangent line.(b) the instantaneous rate of change of the function. y = ( 4 x 3 − 5 x + 1 ) 3  at  ( 1 , 0 )

(a)

To determine

To calculate: The slope of the tangent line for the function, y=(4x35x+1)3, at (1,0).

Explanation

Given Information:

The function is y=(4x3âˆ’5x+1)3 at (1,0).

Formula used:

The power rule, if f(x)=xn, then,

fâ€²(x)=nxnâˆ’1

The property of differentiation, if a function is of the form, g(x)=cf(x), then,

gâ€²(x)=cfâ€²(x)

The property of differentiation, if a function is of the form f(x)=u(x)+v(x), then,

fâ€²(x)=uâ€²(x)+vâ€²(x)

The product rule, if f(x)=u(x)â‹…v(x), then

fâ€²(x)=uâ€²(x)â‹…v(x)+vâ€²(x)â‹…u(x)

The property of differentiation, if a function is of the form y=un, where u=g(x),

dydx=nunâˆ’1dudx

Calculation:

Consider the provided function,

y=(4x3âˆ’5x+1)3

Consider (4x3âˆ’5x+1) to be u,

y=u3

Differentiate both sides with respect to x,

dydx=ddx(u3)

Simplify using the power rule,

dydx=3â‹…u3âˆ’1â‹…dudx=3u2dudx

Substitute (4x3âˆ’5x+1) for u,

dydx=3(4x3âˆ’5

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