Chapter 9.8, Problem 25E

### Mathematical Applications for the ...

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

Chapter
Section

### Mathematical Applications for the ...

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

# If f ( x )   =   16 x 2 — x 3 , what is the rate of change of f ' ( x ) at (1, 15)?

To determine

To calculate: The rate of change of f(x) for the function f(x)=16x2x3 at (1,15).

Explanation

Given Information:

The provided function is f(x)=16x2âˆ’x3 and the point is (1,15).

Formula used:

Power of x rule for function f(x)=xn is fâ€²(x)=nxnâˆ’1, where n is a real number.

Coefficient rule for a constant c is such that, if f(x)=câ‹…u(x), where u(x) is a differentiable function of x, then fâ€²(x)=câ‹…uâ€²(x).

Constant function rule for a constant c is such that, if f(x)=c then fâ€²(x)=0.

Calculation:

Consider the function, f(x)=16x2âˆ’x3

Differentiate the function with respect to x,

fâ€²(x)=ddx(16x2âˆ’x3)=ddx(16x2)âˆ’ddx(x3)=16ddx(x2)âˆ’ddx(x3)

Use the power of x rule,

fâ€²(x)=16(2â‹…x2âˆ’1</

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