Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

After t years the population of a certain town is $P(t) = 50+ 5t$ thousand people. A population $P$ has an associated CO$_2$ level, $C(P) = \frac{\sqrt{P^2 + 1}}{2}$. After $2$ years, the rate at which CO$_2$ level is changing with respect to t will be:

I plugged in (2) into $P(t)$ to get $60$, then plugged $60$ into $C(P)$. Doing this I got $\frac{\sqrt{3601}}{2}$, but that isn't correct, any ideas on what I'm doing wrong?

share|cite|improve this question

They are asking for the rate of change of $C(P)$ not the $C(P)$ after two years ie you have to calculate $$ \frac{dC(P)}{dt}$$

Just put $P(t)$ in $C(P)$ and then differentiate wrt time and put the value of $t=2$ , you will get the answer .

share|cite|improve this answer

Use the chain rule. You want the rate of change of CO$_2$ wrt time. That's $\frac{dC}{dt}$. By the chain rule, $$\frac{dC}{dt} = \frac{dC}{dP}\frac{dP}{dt} = C'(P)P'(t).$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.