Derivative Interpretation?

Question

Suppose C(r) is the total cost of paying off a car loan borrowed at an annual interest rate of r%. What is the practical meaning of C'(r)?

Answer 1

derivative implies the rate at which a function changes at a given point...

if C(r) is a linear function (such as m.r+n) then the derivative, C'(r) is simply constant... implying that the rate of interest has no impact on the total cost...

if it is not linear, then C'(r) must be evaluated to some values.... positive, negative, or zero for some r.....

if C(r) = m.r + n + p r^2 for instance, then
C'(r) = m + 2pr

so we C'(r) is a linear graph showing how fast the cost increases with r.... ideally we would hope for C'(r) = 0 and p>0 so that C(r) is the minimum at r = -m/2p

Answer 2

The derivative means that, at the current interest rate, a minute change in the current interest rate of delta r would change the total cost of the loan by (delta r)*C'(r). For example, if C'(r) was 100 then a change in interest rate by .1 would add approximately 10 dollars to the total cost of the loan.

Answer 3

Derivatives are rates of change relative to whatever variable you have, in your case, r. This is to say, What happens to the dependent variable (C) if I change the independent variable (r)? In this example, C'(r) gives how much change in the cost will result from a little bit of change in the rate. If C(r) is anything but linear (C= mr + b), then this rate of change will depend on what rate you are at. If it IS linear, then your total cost STILL DEPENDS on the rate, but HOW MUCH it depends does not change, no matter what the rate is.

Answer 4

it tells you the slope of the graph of the function C(r)