   Chapter 9.7, Problem 29E 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 = x 2 4 x − 3 4

To determine

To calculate: The simplified form of the derivative of y=x24x34.

Explanation

Given Information:

The function is y=x24x34.

Formula used:

According to the power rule, if f(x)=xn, then,

f(x)=nxn1

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

g(x)=cf(x)

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

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

According to the product rule, if f(x)=u(x)v(x), then

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

The derivative of a constant value, k, is

ddx(k)=0

According to 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=x24x34

Rewrite the function,

y=x2(4x3)14

Consider (4x3) to be u,

y=x2u14

Differentiate both sides with respect to x,

y=ddx(x2u14)

Simplify using the product rule,

y=((ddx(x2))u14+(ddx(u14))x2)

Simplify using the power rule,

y=((2x21)u14+(14u141dudx)x2)=(2xu14+(14u34ddx(u))x2)

Take xu14 common,

y=xu14(2+14u34u14ddx(u)x)=xu14(2+14u

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