   Chapter 9.7, Problem 23E Mathematical Applications for the ...

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

Solutions

Chapter
Section Mathematical Applications for the ...

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

Find the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only. y = ( 2 x − 1 x 2 + x ) 4

To determine

To calculate: The simplified form of the derivative of the function y=(2x1x2+x)4.

Explanation

Given Information:

The provided function is y=(2x1x2+x)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.

Quotient rule for function f(x)=u(x)v(x), where u and v are differentiable functions of x, then f(x)=v(x)u(x)u(x)v(x)[v(x)]2.

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.

Calculation:

Consider the function,

y=(2x1x2+x)4

Consider 2x1x2+x to be u,

y=u4

Differentiate both sides with respect to x,

dydx=ddx(u4)

Use the power rule,

y=4u41dudx=4u3dudx

Substitute 2x1x2+x for u,

y=4(2x1x2+x)3ddx(2x1x2+x)

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