Derivative Question 101

[llewis1]

Beginner struggling with some basics.
Problem:
Suppose C(r) is the total cost of paying off a car loan borrowed at annual interest rate of r%.
[a] What are the units of C'(r)?
What is the practical meaning of C'(r)?
[c] What is its sign?

Answers (???):
[a] ("y per x", right?) So "dollars per annual interest rate"...but that doesn't make sense.
At the specific dollar amount of the loan, the interest rate ____
[c] Posivite (?)

Any Assistance\explanation is appreciated.

 

[soroban]

Hello, llewis1!

Suppose \(\displaystyle C(r)\) is the total cost of paying off a car loan
borrowed at annual interest rate of \(\displaystyle r\%.\)

[a] What are the units of \(\displaystyle C'(r)\)?
What is the practical meaning of \(\displaystyle C'(r)\)?
[c] What is its sign?

[a] The units are "dollars per percent"

It gives the rate at which the cost increases for each 1% increase in interest rate.

[c] Assuming the cost of the loan increases with a higher interest rate,
. . .the derivative is positive.