Which is where? Understanding function and derivatives

[wfws]

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.

[nona.m.nona]

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.

This reasoning sounds sensible to me.

What confuses me is when are what words or terms here represent the FUNCTION and what represents the derivative?

"The length of the contract" is the variable "t". "The monthly payment" is the function "f". "The effect of the length on the payment" is the derivative.