If the total cost function for a manufacturer is given by `C(x)= (500)/(sqrt(2x + 25)) + 5000`, find marginal cost function.

Answer

To find the marginal cost function from the given total cost function \( C(x) = \frac{500}{\sqrt{2x + 25}} + 5000 \), we need to differentiate the cost function with respect to \( x \).

### Step-by-Step Solution:

1. **Identify the Total Cost Function:**
\[ C(x) = \frac{500}{\sqrt{2x + 25}} + 5000 \]

2. **Differentiate the Cost Function:**
To find the marginal cost function \( C'(x) \), we need to differentiate \( C(x) \):
\[ C'(x) = \frac{d}{dx}\left(\frac{500}{\sqrt{2x + 25}}\right) + \frac{d}{dx}(5000) \]
The derivative of a constant (5000) is 0, so we focus on differentiating the first term.

3. **Using the Quotient Rule:**
The first term can be rewritten as \( 500(2x + 25)^{-1/2} \). We will use the chain rule for differentiation:
\[ \frac{d}{dx}\left(500(2x + 25)^{-1/2}\right) = 500 \cdot \left(-\frac{1}{2}(2x + 25)^{-3/2}\right) \cdot \frac{d}{dx}(2x + 25) \]
Here, \( \frac{d}{dx}(2x + 25) = 2 \).

4. **Substituting the Derivative:**
\[ C'(x) = 500 \cdot \left(-\frac{1}{2}(2x + 25)^{-3/2}\right) \cdot 2 \]
Simplifying this gives:
\[ C'(x) = -500(2x + 25)^{-3/2} \]

5. **Final Expression for Marginal Cost:**
Thus, the marginal cost function is:
\[ C'(x) = -\frac{500}{(2x + 25)^{3/2}} \]