Your Porsche's gas mileage (in miles per gallon) is given as a function M(x) of speed x in miles per hour. It is found that

M'(x) =
3,600x−2 − 1/
(3,600x−1 + x)2.

Estimate M'(10), M'(60), and M'(90). (Round your answers to seven decimal places.)
M'(10) = mpg/mph
M'(60) = mpg/mph
M'(90) = mpg/mph

To find the estimates of M'(10), M'(60), and M'(90), we need to evaluate the given derivative function at those specific speeds.

First, let's evaluate M'(10):
Plug the value x = 10 into the derivative function M'(x):
M'(10) = (3,600(10)^-2 - 1) / (3,600(10)^-1 + 10)^2

Simplify the expression inside the parentheses:
M'(10) = (3,600(1/100) - 1) / (3,600(1/10) + 10)^2

Calculate the values inside the parentheses:
M'(10) = (36 - 1) / (360 + 10)^2

Simplify further:
M'(10) = 35 / (370)^2

Evaluate the expression:
M'(10) ≈ 35 / 136,900 ≈ 0.000255485

Next, let's evaluate M'(60):
Plug the value x = 60 into the derivative function M'(x):
M'(60) = (3,600(60)^-2 - 1) / (3,600(60)^-1 + 60)^2

Simplify the expression inside the parentheses:
M'(60) = (3,600(1/3600) - 1) / (3,600(1/60) + 60)^2

Calculate the values inside the parentheses:
M'(60) = (1 - 1) / (1/60 + 60)^2

Simplify further:
M'(60) = 0 / (1/60 + 60)^2

Evaluate the expression:
M'(60) = 0 / (1/60 + 60)^2 = 0

Finally, let's evaluate M'(90):
Plug the value x = 90 into the derivative function M'(x):
M'(90) = (3,600(90)^-2 - 1) / (3,600(90)^-1 + 90)^2

Simplify the expression inside the parentheses:
M'(90) = (3,600(1/8100) - 1) / (3,600(1/90) + 90)^2

Calculate the values inside the parentheses:
M'(90) = (1/2 - 1) / (1/90 + 90)^2

Simplify further:
M'(90) = (-1/2) / (1/90 + 90)^2

Evaluate the expression:
M'(90) ≈ (-1/2) / (1/90 + 90)^2 ≈ -0.000012346

Therefore, the estimated values are:
M'(10) ≈ 0.000255485
M'(60) = 0
M'(90) ≈ -0.000012346

(Note: The values are rounded to seven decimal places as requested.)

To estimate M'(10), M'(60), and M'(90), we need to evaluate the given derivative function at those specific speeds.

1. M'(10):
Substitute x = 10 into the derivative function:
M'(10) = (3,600(10)^(-2) - 1) / (3,600(10)^(-1) + 10)^2

Simplifying the expression:
M'(10) = (3,600 * 0.01 - 1) / (3,600 * 0.1 + 10)^2
M'(10) = (36 - 1) / (360 + 10)^2
M'(10) = 35 / (370)^2
M'(10) ≈ 0.0026832239 (rounded to seven decimal places)

2. M'(60):
Substitute x = 60 into the derivative function:
M'(60) = (3,600(60)^(-2) - 1) / (3,600(60)^(-1) + 60)^2

Simplifying the expression:
M'(60) = (3,600 * 0.00027778 - 1) / (3,600 * 0.01667 + 60)^2
M'(60) = (1 - 1) / (60.6)^2
M'(60) ≈ 0 (rounded to seven decimal places)

3. M'(90):
Substitute x = 90 into the derivative function:
M'(90) = (3,600(90)^(-2) - 1) / (3,600(90)^(-1) + 90)^2

Simplifying the expression:
M'(90) = (3,600 * 0.00012346 - 1) / (3,600 * 0.01111 + 90)^2
M'(90) = (-0.999999 - 1) / (144.9)^2
M'(90) ≈ -0.001507048 (rounded to seven decimal places)

Estimates:
M'(10) ≈ 0.0026832239 mpg/mph
M'(60) ≈ 0 mpg/mph
M'(90) ≈ -0.001507048 mpg/mph