Find the slope of average cost curve for the total cost function `C=ax^(3)+bx^(2)+cx +d.`

Answer

To find the slope of the average cost curve for the total cost function \( C(x) = ax^3 + bx^2 + cx + d \), we can follow these steps:

### Step 1: Define the Average Cost Function
The average cost \( AC \) is defined as the total cost \( C(x) \) divided by the quantity \( x \):
\[ AC(x) = \frac{C(x)}{x} = \frac{ax^3 + bx^2 + cx + d}{x} \]

### Step 2: Simplify the Average Cost Function
We can simplify the average cost function:
\[ AC(x) = \frac{ax^3}{x} + \frac{bx^2}{x} + \frac{cx}{x} + \frac{d}{x} = ax^2 + bx + c + \frac{d}{x} \]

### Step 3: Differentiate the Average Cost Function
To find the slope of the average cost curve, we need to differentiate \( AC(x) \) with respect to \( x \):
\[ \frac{d(AC)}{dx} = \frac{d}{dx}(ax^2 + bx + c + \frac{d}{x}) \]

### Step 4: Apply Differentiation Rules
Using the power rule for differentiation, we differentiate each term:
- The derivative of \( ax^2 \) is \( 2ax \).
- The derivative of \( bx \) is \( b \).
- The derivative of \( c \) (a constant) is \( 0 \).
- The derivative of \( \frac{d}{x} \) is \( -\frac{d}{x^2} \) (using the power rule for \( x^{-1} \)).

Putting it all together:
\[ \frac{d(AC)}{dx} = 2ax + b - \frac{d}{x^2} \]

### Step 5: Final Expression for the Slope of Average Cost
Thus, the slope of the average cost curve is:
\[ \frac{d(AC)}{dx} = 2ax + b - \frac{d}{x^2} \]