Differentiate the following functions:
`sqrt( x ) - (1)/( sqrt x) , x ne 0`

Answer

To differentiate the function \( f(x) = \sqrt{x} - \frac{1}{\sqrt{x}} \), we will follow these steps:

### Step 1: Rewrite the function
We can express the square root and the reciprocal in terms of exponents:
\[ f(x) = x^{1/2} - x^{-1/2} \]

### Step 2: Differentiate using the power rule
Using the power rule for differentiation, which states that \( \frac{d}{dx} x^m = m \cdot x^{m-1} \), we differentiate each term:
1. Differentiate \( x^{1/2} \):
\[ \frac{d}{dx}(x^{1/2}) = \frac{1}{2} x^{1/2 - 1} = \frac{1}{2} x^{-1/2} \]
2. Differentiate \( -x^{-1/2} \):
\[ \frac{d}{dx}(-x^{-1/2}) = -\left(-\frac{1}{2} x^{-1/2 - 1}\right) = \frac{1}{2} x^{-3/2} \]

### Step 3: Combine the derivatives
Now, we combine the derivatives from both terms:
\[ f'(x) = \frac{1}{2} x^{-1/2} + \frac{1}{2} x^{-3/2} \]

### Step 4: Simplify the expression
We can factor out \( \frac{1}{2} \):
\[ f'(x) = \frac{1}{2} \left( x^{-1/2} + x^{-3/2} \right) \]

### Step 5: Rewrite in terms of square roots
Rewriting the expression in terms of square roots gives:
\[ f'(x) = \frac{1}{2} \left( \frac{1}{\sqrt{x}} + \frac{1}{x\sqrt{x}} \right) \]

### Step 6: Combine into a single fraction
Finding a common denominator:
\[ f'(x) = \frac{1}{2} \left( \frac{x + 1}{x\sqrt{x}} \right) \]

### Final Answer
Thus, the derivative of the function \( f(x) = \sqrt{x} - \frac{1}{\sqrt{x}} \) is:
\[ f'(x) = \frac{x + 1}{2x\sqrt{x}}, \quad x \neq 0 \]

---