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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?

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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 .

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

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