If the total cost function for a manufacturer is given by `C=(5x^(2))/(sqrt(x^(2)+3))+500`. Find the marginal cost.

Answer

To find the marginal cost from the given total cost function \( C = \frac{5x^2}{\sqrt{x^2 + 3}} + 500 \), we need to differentiate the cost function with respect to \( x \).

### Step-by-Step Solution:

1. **Identify the Total Cost Function**:
\[ C = \frac{5x^2}{\sqrt{x^2 + 3}} + 500 \]

2. **Differentiate the Cost Function**:
The marginal cost \( MC \) is given by the derivative of \( C \) with respect to \( x \):
\[ MC = \frac{dC}{dx} \]

3. **Apply the Quotient Rule**:
The quotient rule states that if you have a function \( \frac{u}{v} \), then its derivative is given by:
\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} \]
Here, \( u = 5x^2 \) and \( v = \sqrt{x^2 + 3} \).

- Differentiate \( u \):
\[ u' = \frac{d}{dx}(5x^2) = 10x \]
- Differentiate \( v \):
\[ v = (x^2 + 3)^{1/2} \implies v' = \frac{1}{2}(x^2 + 3)^{-1/2} \cdot 2x = \frac{x}{\sqrt{x^2 + 3}} \]

4. **Substitute into the Quotient Rule**:
Now, substituting \( u, u', v, \) and \( v' \) into the quotient rule:
\[ MC = \frac{(10x)(\sqrt{x^2 + 3}) - (5x^2)\left(\frac{x}{\sqrt{x^2 + 3}}\right)}{(x^2 + 3)} \]

5. **Simplify the Expression**:
- The numerator becomes:
\[ 10x\sqrt{x^2 + 3} - \frac{5x^3}{\sqrt{x^2 + 3}} \]
- To combine these terms, multiply the first term by \( \sqrt{x^2 + 3} \):
\[ \frac{10x(x^2 + 3) - 5x^3}{\sqrt{x^2 + 3}} = \frac{10x^3 + 30x - 5x^3}{\sqrt{x^2 + 3}} = \frac{5x^3 + 30x}{\sqrt{x^2 + 3}} \]

6. **Final Expression for Marginal Cost**:
Now, we can write the marginal cost as:
\[ MC = \frac{5x(x^2 + 6)}{(x^2 + 3)^{3/2}} \]

### Final Answer:
\[ MC = \frac{5x(x^2 + 6)}{(x^2 + 3)^{3/2}} \]