a software company estimates

Question

A software company estimates that the number of copies x, measured in thousands, of its new application that will sell when the price is p dollars per copy (up to $100) satisfies the equation 2px² + 3px = 58,000.

 Find dx/dp

 when the price is $40 per copy. (Round your answer to three decimal places.)

When the price increases from $40, the number of copies that will sell decreases at a rate of

Answer 1

Differentiate the entire expression with respect to p (we'll be using implicit differentiation)

d/dp (2px² + 3px = 58,000) (Equ.1)

let's evaluate each term, from right to left. let dx/dp=x'

d(58000)/dp = 0, constant

d(3px)/dp = 3x + 3p*x' , , chain rule/multiplication rule

d/dp (2px²) = 2x² + 2p*2x*x' = 2x² + 4px*x'

Turning our first equation (Equ.1) into

2x² + 4px*x' + 3x + 3p*x' =0

solve for x'

x'(3p + 4px) = -2x² -3x

x' = -(2x² -3x)/(3p + 4px) (Equ.2)

we can plug in p=40 into our original equation (Equ.1) to find the value of x

2(40)x² + 3(40)x = 58,000

80x² + 120x = 58,000, use the quadratic equation to find the value of x

→ x ≈ 26.1863

Finally, you can plug in p=40 and x≈26.1863 into our x' (Equ.2) to get the answer

x' ≈ -.2999 ≈ -.300

When p=$40 per unit, the number of projected sold copies decreases at a rate of .3 per unit price