Answer
(a) $f'(r)$ is the instantaneous rate of change of the total cost of repaying the loan at an interest rate of $r$
The units of $f'(r)$ are $\$/(percent~per~year)$
(b) $f'(10) = 1200$ means that at an interest rate of 10% per year, the total cost of the loan is increasing at a rate of $\$1200/(percent~per~year)$
(c) $f'(r)$ is always positive.
Work Step by Step
(a) The interest rate is $r$% per year
The total cost of repaying a student loan is $C = f(r)$
$f'(r)$ is the instantaneous rate of change of the total cost of repaying the loan at an interest rate of $r$
The units of $f'(r)$ are $\$/(percent~per~year)$
(b) Let's suppose that the interest rate $r$ is increasing.
$f'(10) = 1200$ means that when the interest rate reaches 10% per year, the total cost of the loan is increasing at a rate of $\$1200/(percent~per~year)$
(c) Since a higher interest rate always results in a higher total cost to repay the loan, $f'(r)$ is always positive.