   Chapter 9.7, Problem 21E ### 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

# Find the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only. R ( x ) = [ x 2 ( x 2 + 3 x ) ] 4

To determine

To calculate: The simplified form of the derivative of the function R(x)=[x2(x2+3x)]4.

Explanation

Given Information:

The provided function is R(x)=[x2(x2+3x)]4.

Formula used:

Power rule for a real number n is such that, if y=un then dydx=nun1dudx, where u is a differentiable function of x.

Product rule for function f(x)=u(x)v(x), where u and v are differentiable functions of x, then f(x)=u(x)v(x)+v(x)u(x).

Power of x rule for function f(x)=xn is f(x)=nxn1, where n is a real number.

Coefficient rule for a constant c is such that, if f(x)=cu(x), where u(x) is a differentiable function of x, then f(x)=cu(x).

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

The power rule for exponent is such that, (am)n=amn, where a is a real number and m, n are integers.

Calculation:

Consider the function,

R(x)=[x2(x2+3x)]4

Consider x2(x2+3x) to be u,

R(x)=u4

Differentiate both sides with respect to x,

R(x)=ddx(u4)

Use the power rule,

R(x)=4u41dudx=4u3dudx

Substitute x2(x2+3x) for u,

R(x)=4

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