I have an exercise that I sorta get, but I need some help to be sure I'm getting it right. The problem is about determining the effect of the length of a contract on a monthly payment for a car purchase. The price and interest rate are fixed.

P = f(t) is the monthly payment in $, and the contract length is t months.

Which value doesn't make sense: f'(36) = 12 or f'(36) = -12?

I said that f'(36)= 12 doesn't make sense because it indicates that a longer length term would increase payments ( 12 would indicate a positive slope on the function). Since payments should decrease (slope downwards) as contract lenth in months is longer, the number should be negative, not positive.

Am I understanding this correctly? What confuses me is when are what words or terms here represent the FUNCTION and what represents the derivative?

Thanks for any insights! I have several more and I want to get this concept.