Given that `y=x^(2)+4x^(-(1)/(2))-3x^(-2),` find `(dy)/(dx)`.

Options

A. `2x-2x^(-(3)/(2))+6x^(-3)`
B. `-2x-2x^(-(3)/(2))+6x^(-3)`
C. `2x-2x^(-(1)/(2))+6x^(-3)`
D. `=-2x-2x^(-(3)/(2))+6x^(-2)`

Explanation

To find the derivative \( \frac{dy}{dx} \) for the function \( y = x^2 + 4x^{-\frac{1}{2}} - 3x^{-2} \), we will differentiate each term separately using the power rule.

### Step-by-Step Solution:

1. **Identify the function**:
\[ y = x^2 + 4x^{-\frac{1}{2}} - 3x^{-2} \]

2. **Differentiate the first term \( x^2 \)**:
Using the power rule \( \frac{d}{dx}(x^n) = nx^{n-1} \):
\[ \frac{d}{dx}(x^2) = 2x^{2-1} = 2x \]

3. **Differentiate the second term \( 4x^{-\frac{1}{2}} \)**:
Again applying the power rule:
\[ \frac{d}{dx}(4x^{-\frac{1}{2}}) = 4 \cdot \left(-\frac{1}{2}\right)x^{-\frac{1}{2}-1} = -2x^{-\frac{3}{2}} \]

4. **Differentiate the third term \( -3x^{-2} \)**:
Applying the power rule:
\[ \frac{d}{dx}(-3x^{-2}) = -3 \cdot (-2)x^{-2-1} = 6x^{-3} \]

5. **Combine all the derivatives**:
Now, we can combine all the differentiated terms:
\[ \frac{dy}{dx} = 2x - 2x^{-\frac{3}{2}} + 6x^{-3} \]

6. **Final expression**:
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 2x - 2x^{-\frac{3}{2}} + 6x^{-3} \]