The derivative of `f(x) = (sqrt(x) + (1)/(sqrt(x)))^(2)` is

Options

A. `sqrt(x) + (1)/(sqrt(x))`
B. `2 ((sqrt(x) + 1)/(sqrt(x)))`
C. `1 - (1)/(x^(2))`
D. `1 + (1)/(x^(2))`

Answer

To find the derivative of the function \( f(x) = \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^2 \), we can follow these steps:

### Step 1: Rewrite the function
We start with the function:
\[ f(x) = \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^2 \]

### Step 2: Apply the square expansion
Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \), we can expand the function:
\[ f(x) = \left( \sqrt{x} \right)^2 + 2 \left( \sqrt{x} \right) \left( \frac{1}{\sqrt{x}} \right) + \left( \frac{1}{\sqrt{x}} \right)^2 \]
This simplifies to:
\[ f(x) = x + 2 + \frac{1}{x} \]

### Step 3: Differentiate the function
Now we differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x) + \frac{d}{dx}(2) + \frac{d}{dx}\left(\frac{1}{x}\right) \]
Calculating each term:
- The derivative of \( x \) is \( 1 \).
- The derivative of \( 2 \) is \( 0 \).
- The derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \).

Putting it all together:
\[ f'(x) = 1 + 0 - \frac{1}{x^2} \]
Thus, we have:
\[ f'(x) = 1 - \frac{1}{x^2} \]

### Final Answer
The derivative of \( f(x) \) is:
\[ f'(x) = 1 - \frac{1}{x^2} \]

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