The cost, in dollars, of producing x belts is given by Upper C (x )equals 594 plus 10 x minus 0.067 x squared. Find the rate at which average cost is changing when 576 belts have been produced. When 576 belts have been produced, the average cost is changing at nothing dollars per belt belts per dollar for each additional belt.

To find the rate of average cost that is, [tex] \dfrac{d\overline{C}}{dx}[/tex]. So differentiating function [tex] \overline{C}[/tex] with respect to x,

[tex] \dfrac{d\overline{C}}{dx}=\dfrac{d}{dx}\left (\dfrac{594}{x}+10-0.067x \right )[/tex]

Applying sum rule and difference rule of derivative,

[tex] \dfrac{d\overline{C}}{dx}=\dfrac{d}{dx}\left(\dfrac{594}{x}\right)+\dfrac{d}{dx}\left(10\right)-\dfrac{d}{dx}\left(0.067x\right)[/tex]

Applying constant multiple rule of derivative,

[tex] \dfrac{d\overline{C}}{dx}=594\dfrac{d}{dx}\left(\dfrac{1}{x}\right)+\dfrac{d}{dx}\left(10\right)-0.067\dfrac{d}{dx}\left(x\right)[/tex]

Since,

[tex] \dfrac{1}{x}=x^{-1}[/tex]

[tex] \dfrac{d\overline{C}}{dx}=594\dfrac{d}{dx}\left(x^{-1}\right)+\dfrac{d}{dx}\left(10\right)-0.067\dfrac{d}{dx}\left(x\right)[/tex]

Applying power rule and constant rule of derivative,

[tex] \dfrac{d\overline{C}}{dx}=594\left(-1\:x^{-1-1}\right)+0-0.067\left(1\:x^{1-1}\right)[/tex]

[tex] \dfrac{d\overline{C}}{dx}=594\left(-1\:x^{-2}\right)-0.067\left(1\:x^{0}\right) [/tex]

[tex] \dfrac{d\overline{C}}{dx}=594\:x^{-2}-0.067 [/tex]

[tex]\dfrac{d\overline{C}}{dx}=\dfrac{594}{x^{2}}-0.067[/tex]

Substituting the value of [tex]x=576[/tex],

[tex]\dfrac{d\overline{C}}{dx}=594\left(\dfrac{1}{\left(576 \right)^{2}}\right )-0.067[/tex]

[tex] \dfrac{d\overline{C}}{dx}=-0.0652 [/tex]

Negative sign indicates that rate of average cost is decreasing.

Therefore, the average cost is changing at the rate of 0.0652(decreasing) dollar per belt