If the total cost function is given by C(x), where x is the quality of the output, then `(d)/(dx)(AC)` =

Options

A. `(1)/(x)(AC - MC)`
B. `(d)/(dx) (MC)`
C. `(1)/(x^(2)) (MC - AC)`
D. None of these

Answer

To solve the problem, we need to find the derivative of the Average Cost (AC) function with respect to the quantity of output (x). Let's go through the steps to derive this.

### Step 1: Understand the Average Cost Function
The Average Cost (AC) is defined as the total cost (C) divided by the quantity of output (x):
\[ AC = \frac{C(x)}{x} \]

### Step 2: Differentiate the Average Cost Function
To find \(\frac{d}{dx}(AC)\), we will use the quotient rule of differentiation. The quotient rule states that if you have a function \( \frac{u}{v} \), then its derivative is given by:
\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]
In our case, \(u = C(x)\) and \(v = x\).

### Step 3: Apply the Quotient Rule
Now we apply the quotient rule:
\[ \frac{d}{dx}(AC) = \frac{x \cdot C'(x) - C(x) \cdot 1}{x^2} \]
Where \(C'(x)\) is the marginal cost (MC).

### Step 4: Simplify the Expression
This simplifies to:
\[ \frac{d}{dx}(AC) = \frac{x \cdot MC - C(x)}{x^2} \]

### Step 5: Relate to Marginal Cost and Average Cost
We can express this in terms of AC:
\[ \frac{d}{dx}(AC) = \frac{MC - AC}{x} \]

### Conclusion
Thus, the final result for the derivative of the Average Cost with respect to output is:
\[ \frac{d}{dx}(AC) = \frac{MC - AC}{x} \]