Question

Suppose that the cost of producing n pairs of shoes is given by the expression c(n)=30+5n. the average cost of producing a pair of shoes is given by a(n)=c(n)/n. what is the derivative of the average cost function at n=10?

Answer 1

Given

$c(n)=30+5n\\a(n)=\dfrac{c(n)}{n}$

Find

$\dfrac{d}{dn}(a(n))\quad\text{at n=10}$

Solution

$\dfrac{d}{dn}(a(n))=\dfrac{d}{dn}\left(\dfrac{30+5n}{n}\right)=\dfrac{d}{dn}\left(30n^{-1}+5\right)\\\\=-30n^{-2}\\\\=-30\cdot 10^{-2}\qquad\text{at n=10}\\\\=-0.30$

The derivative of the average cost function at n=10 is -0.30.