Chain rule for P(n) = 0.25 sqrt(0.5 n^2 + 5n + 25)

[AGlas9837]

An environmental study indicates that the average daily level P of a certain pollutant in the air in parts per million can be modeled by the equation: P = 0.25 times the square root of (0.5n^2 + 5n + 25) where n is the number of residents of the community in thousands. Find the rate at which the level of pollutant is increasing when the population of the community is 12,000.

Heres' what I've got:

P' = 0.25 (.5n^2 + 5n + 25) to the 1/2 power
P' = (0.25) (1/2) (.5n^2 + 5n + 25) to the -1/2 power (.10n + 5) + (.5n^2 + 5n + 25) to the 1/2 power (0)
P' = (0.25) (1/2) (.5n^2 + 5n + 25) to the -1/2 power (.10n + 5)

From here, it doesn't look like I can factor anything. Do I even have this much correct?

 

[stapel]

P' = 0.25 (.5n^2 + 5n + 25) to the 1/2 power

P' = (0.25) (1/2) (.5n^2 + 5n + 25) to the -1/2 power (.10n + 5)

From here, it doesn't look like I can factor anything. Do I even have this much correct?

Why would you need to factor? Aren't you supposed to be evaluating at n = 12,000...?

Also, I'm fairly certain that (1/2)(2/1) = 1, not one-tenth, so you might want to check your derivative of the argument of the radical.

Eliz.

 

[pka]

\(\displaystyle \begin{array}{l} P(n) = .25\sqrt {0.5n^2 + 5n + 25} \\ P'(n) = \frac{{.25\left( {n + 5} \right)}}{{2\sqrt {0.5n^2 + 5n + 25} }} \\ \end{array}\)

 

[AGlas9837]

Thank you pka. Except for the multiplication error, I had most of the problem worked out correctly and I understand perfectly now...just didn't see it before.