   Chapter 11.1, Problem 47E ### 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

# Demand The demand function for a product is given by p = 4000/ln (x + 10), where p is the price per unit in dollars when x units are demanded.(a) Find the rate of change of price with respect to the number of units sold when 40 units are sold.(b) Find the rate of change of price with respect to the number of units sold when 90 units are sold.(c) Find the second derivative to see whether the rate at which the price is changing at 40 units is increasing or decreasing.

(a)

To determine

To calculate: The rate of change of price with respect to the number of units sold for 40 units sold.

Explanation

Given Information:

The demand function for a product is given by,

p=4000ln(x+10)

Where p is the price of unit in dollars when x units are demanded.

Formula Used:

According to the property of derivative,

ddx(lnx)=1x

By quotient rule of derivatives,

ddx(uv)=uvuvu2

Chain rule,

ddxf(g(x))=fg(x)g(x)

Calculation:

Consider the provided function p=4000ln(x+10).

The rate of change of the price with respect to the number of units sold is given by,

p(x)=ddx(p(x))=ddx(4000ln(x+10))

Apply the quotient rule of derivative to get,

p(x)=ddx(p(x))=ddx(4000ln(x+10))=ddx(4000)ln(x+10)(4000)ddx(ln(x+10))(ln(x+10))2

Apply the chain rule to get,

p(x)=ddx(4000)ln(x+10)(4000)ddx(ln(x+10))(ln(x+10))2

(b)

To determine

To calculate: The rate of change of price with respect to the number of units sold for 90 units sold.

(c)

To determine

To calculate: The second derivative to see whether the rate at which the price is changing at 40 units is increasing or decreasing.

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