   Chapter 9.6, Problem 25E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
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 = x 3 − 1  at  ( 2 , 3 )

(a)

To determine

To calculate: The slope of the tangent line for the function, y=x3+1, at (2,3).

Explanation

Given Information:

The function is y=x3+1, at (2,3).

Formula used:

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

f(x)=nxn1

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

g(x)=cf(x)

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

f(x)=u(x)+v(x)

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

f(x)=u(x)v(x)+v(x)u(x)

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

dydx=nun1dudx

Calculation:

Consider the provided function,

y=x3+1

Rewrite the function,

y=(x3+1)12

Consider (x3+1) to be u,

y=u12

Differentiate both sides with respect to x,

dydx=ddx(u12)

Simplify using the power rule,

dydx=12u121dudx=12u122

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