If `C(x)=200x-5x^(2)+(x^(2))/(3)`, then find the marginal cost (MC)

Answer

To find the marginal cost (MC) from the given cost function \( C(x) = 200x - 5x^2 + \frac{x^2}{3} \), we need to compute the derivative of the cost function with respect to \( x \).

### Step-by-Step Solution:

1. **Identify the Cost Function**:
\[ C(x) = 200x - 5x^2 + \frac{x^2}{3} \]

2. **Differentiate the Cost Function**:
To find the marginal cost, we differentiate \( C(x) \) with respect to \( x \):
\[ \frac{dC}{dx} = \frac{d}{dx}(200x) - \frac{d}{dx}(5x^2) + \frac{d}{dx}\left(\frac{x^2}{3}\right) \]

3. **Calculate Each Derivative**:
- The derivative of \( 200x \) is \( 200 \).
- The derivative of \( -5x^2 \) is \( -10x \) (using the power rule).
- The derivative of \( \frac{x^2}{3} \) is \( \frac{2x}{3} \) (again using the power rule).

4. **Combine the Derivatives**:
Now, we combine these derivatives:
\[ \frac{dC}{dx} = 200 - 10x + \frac{2x}{3} \]

5. **Simplify the Expression**:
To combine \( -10x \) and \( \frac{2x}{3} \), we can express \( -10x \) as \( -\frac{30x}{3} \):
\[ \frac{dC}{dx} = 200 - \frac{30x}{3} + \frac{2x}{3} = 200 - \frac{30x - 2x}{3} = 200 - \frac{28x}{3} \]

6. **Final Expression for Marginal Cost**:
Thus, the marginal cost function \( MC \) is:
\[ MC = 200 - \frac{28x}{3} \]

### Final Answer:
\[ MC = 200 - \frac{28x}{3} \]