`(d)/(dx)(sqrtx+(1)/(sqrtx))^2=`

Options

A. `1-(1)/(x^2)`
B. `1+(1)/(x^2)`
C. `1-(1)/(2x)`
D. none of these

Explanation

To solve the problem \(\frac{d}{dx}\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2\), we will follow these steps:

### Step 1: Rewrite the function
Let \( f(x) = \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 \).

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

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

So,
\[ \frac{d}{dx} f(x) = 1 - \frac{1}{x^2} + 0 \]
Thus,
\[ \frac{d}{dx} f(x) = 1 - \frac{1}{x^2} \]

### Final Answer
The derivative is:
\[ \frac{d}{dx}\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 = 1 - \frac{1}{x^2} \]