I have a problem here relating to derivatives:

Question

An accessories company finds that the​ cost, in​ dollars, of producing x belts is given by C(x) = 740 + 30x - 0.068x^2

Find the rate at which average cost is changing when 171 bells have been produced. First find the rate at which the average cost is changing when x belts have been produced. barC'(x) = ?

The question it wants me to fill in is what is confusing me, because in the C'(x) = ?, there is a small bar right above the C. What does that little bar mean?

Answer 1

The bar above the letter "C" indicates that it is the average cost, denoted by "C-bar" or "barC". When you take the derivative of the average cost function, you get the rate at which the average cost is changing with respect to the number of belts produced. Therefore, the expression you're looking for is "barC'(x)".

Answer 2

It simply means to take the derivative of C(x), also stated as
d C(x)/dx or simply C'(x)

C'(x) = 30 - .136x
so C'(171) = 30 - .136(171) = $6.744

Answer 3

C(x) is the total cost, so the average cost is
C̅(x) = C(x)/x = 740/x + 30 - 0.068x
C̅'(x) = -740/x^2 - 0.068
so now we have
C̅'(171) = -0.0933