Meaning of derivative in terms of interest rate?

[hvidales]

Homework Statement

The total cost of repaying a student loan at an interest rate of r% per year is C=f(r).
a. What is the meaning of the derivative f '(r)?
b. What does the statement f '(10)=1200 mean?

Homework Equations

The Attempt at a Solution

For part a I think the meaning of the derivative f '(r) means change in interest rate and for part b I think that f '(10) is the interest rate and 1200 is the cost.
I am not sure though so does anybody know if I am correct? Thank you in advance!

 

[Pengwuino]

Close. When you're talking about derivatives, you're talking about the change in one thing as you change some other thing. So you have two things, the total cost of paying back the loan and the interest rate. The derivative you have is the change in the total cost of paying back the loan as you vary the interest rate! If this is a realistic function, you expect that as you increase the interest rate, the total cost to pay back the loan will increase as well. The derivative will tell you that relationship of how quickly the total cost will change as your rate changes.

So given f'(10) = 1200, what that says is that around 10% interest rate, the rate of change of the total cost to pay off the loan is 1200 dollars/%. What that means is that as long as the function doesn't vary wildly, one would expect by raising the interest rate from 10% to 11%, the cost of the loan in the end would be 1200 MORE.

Of course, how accurate that is depends on how much you change the interest rate by (recall that the derivative at a point shows the slope at that point only).

 

[hvidales]

I see and now that makes so much sense. Thank you very much Pengwuino!