Answer
the population of a certain collection of rare Brazilian ants is given by
$$
\begin{aligned}
P(t)&=(t+100)\ln(t+2),
\end{aligned}
$$
The rates of change of the population on the second day and on the eighth day.
(*) the second day :
$$
\begin{aligned}
P^{\prime}(2) \approx 26.9
\end{aligned}
$$
(*) the eighth day:
$$
\begin{aligned}
P^{\prime}(8) \approx 13.1
\end{aligned}
$$
Work Step by Step
the population of a certain collection of rare Brazilian ants is given by
$$
\begin{aligned}
P(t)&=(t+100)\ln(t+2),
\end{aligned}
$$
the rates of change of the population respect to $t$
$$
\begin{aligned}
P^{\prime}(t)&=(t+100). \left[ \frac{1}{t+2}\right] + \ln(t+2).
\end{aligned}
$$
The rates of change of the population on the second day and on the eighth day.
(*) the second day :
$$
\begin{aligned}
P^{\prime}(2)&=(2+100). \left[ \frac{1}{2+2}\right] + \ln(2+2)\\
&=(102). \left[ \frac{1}{4}\right] + \ln(4)\\
& \approx 26.9
\end{aligned}
$$
(*) the eighth day:
$$
\begin{aligned}
P^{\prime}(8)&=(8+100). \left[ \frac{1}{8+2}\right] + \ln(8+2)\\
&=(108). \left[ \frac{1}{10}\right] + \ln(10)\\
& \approx 13.1
\end{aligned}
$$
