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

I will read that as

M'(x) = (3600x^-2 - 1)/(3600x^-1 + x)^2

which is
= (3600/x^2 - 1)/(3600/x + x)^2

M'(10) = (3600/100 - 1)/(3600/10+ 10)^2
= 35/(370)^2
= ????

This makes no sense
Either I did not interpret your typing correctly, or there is a typo.
Please use ^ to exponents and brackets to establish where the division is.
e.g. is it
(3600x^-2 - 1)/(3600x^-1 + x)^2
or
3600x^-2 - ( 1/(3600x^-1 + x)^2 ) ?

ok, maybe this is more accurate...

M'(x) = (3600x^-2 -1)/
(3600x^-1 +x)^2

you plug in 10,60,90 for x

so
3600(10)^-1 -1
/
(3600(10)^2 +10)^2 = 0.0002557 round to the seven decimal places also and you do that for each one.

To estimate M'(10), M'(60), and M'(90), we need to evaluate the derivative function M'(x) at those values of x.

To calculate the derivative of M(x), we use the rules of differentiation. In this case, the function M(x) is given as M(x) = 1 / (3,600x + x^2).

Now let's find M'(x):

M'(x) = [d/dx (1)] / (3,600x+x^2) + 1 / [d/dx (3,600x+x^2)]
= 0 / (3,600x+x^2) + 1 / (3,600 + 2x)

Now we can substitute the values x = 10, x = 60, and x = 90 into the derivative function M'(x) to estimate M'(10), M'(60), and M'(90).

M'(10) = 1 / (3,600 + 2*10)
= 1 / (3,600 + 20)
= 1 / 3,620
≈ 0.000276243

M'(60) = 1 / (3,600 + 2*60)
= 1 / (3,600 + 120)
= 1 / 3,720
≈ 0.000268817

M'(90) = 1 / (3,600 + 2*90)
= 1 / (3,600 + 180)
= 1 / 3,780
≈ 0.000264901

Therefore, the estimated values for M'(10), M'(60), and M'(90) are approximately 0.000276243 mpg/mph, 0.000268817 mpg/mph, and 0.000264901 mpg/mph, respectively.