This confused me more than it should have, thinking through "is the interest rate really the time derivative of the amount of money outstanding on my loan?" It

This confused me more than it should have, thinking through "is the interest rate really the time derivative of the amount of money outstanding on my loan?" It might be easier to think of it in terms of a bank deposit. If I put $100 in a bank account and the rate of interest is 5%/year, after a year my balance has grown by $5. Of course, a loan works the exact same way in reverse. But as consumers normally we think of a loan as being paid off over a fixed length of time, so the rate of change is the rate at which you pay it off, and only second-order influenced by the interest rate.