Let C(x) be the cost of producing x cars in a factory...(derivatives inside)

Hey 

Let C(x) be the cost of producing cars in a factory

Find the formula A(x), the average price of producing a car in the factory when x cars are being produced 

Let C(1)=100$ 

C(2)=200$ 

C(3)=300$... 

To find the average all we need to do is add 2 prices then divide by 2 

so 100$+200$=300$

300$/2=150$

The average price is 150$ for one car 

So A(1)=150$ 

for A(2)=200$+300$=500$/2=250$

Hence the ratio of change is 100$ 

Now to write A(x) in terms of x 

A(x)=((C(x)+C(x+1)/2 

To test the function we use C(1) and C(2) 

A(1)=((C(1)+C(2))/2  (100+200)/2=150

So it works! 

Now to derive We know A(x)=((C(x)+C(x+1))/2 

So we derive that, after deriving that and simplify we Get \(d/dx(1/2 (C(x)+C(x+1))) = 1/2 (C'(x)+C'(x+1)) \) 

Now deriving C(x) we get \(C'(x)\) Sorry if i got some of the derivatives wrong