If `y=sqrt(x)+(1)/(sqrt(x))`, then `(dy)/(dx)` at `x=1` is

Options

A. 1
B. `1/2`
C. `1/(sqrt(2))`
D. 0

Answer

To find \(\frac{dy}{dx}\) for the function \(y = \sqrt{x} + \frac{1}{\sqrt{x}}\) at \(x = 1\), we will follow these steps:

### Step 1: Differentiate \(y\)
Given:
\[ y = \sqrt{x} + \frac{1}{\sqrt{x}} \]

We need to differentiate \(y\) with respect to \(x\).

The derivative of \(\sqrt{x}\) is:
\[ \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \]

The derivative of \(\frac{1}{\sqrt{x}}\) can be rewritten as \(x^{-1/2}\):
\[ \frac{d}{dx}\left(\frac{1}{\sqrt{x}}\right) = \frac{d}{dx}(x^{-1/2}) = -\frac{1}{2}x^{-3/2} = -\frac{1}{2\sqrt{x^3}} \]

Combining these, we have:
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{x}} - \frac{1}{2\sqrt{x^3}} \]

### Step 2: Simplify \(\frac{dy}{dx}\)
We can factor out \(\frac{1}{2}\):
\[ \frac{dy}{dx} = \frac{1}{2}\left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x^3}}\right) \]

### Step 3: Substitute \(x = 1\)
Now, we will evaluate \(\frac{dy}{dx}\) at \(x = 1\):
\[ \frac{dy}{dx}\bigg|_{x=1} = \frac{1}{2}\left(\frac{1}{\sqrt{1}} - \frac{1}{\sqrt{1^3}}\right) \]

Calculating the terms:
\[ \frac{1}{\sqrt{1}} = 1 \quad \text{and} \quad \frac{1}{\sqrt{1^3}} = 1 \]

Thus,
\[ \frac{dy}{dx}\bigg|_{x=1} = \frac{1}{2}(1 - 1) = \frac{1}{2}(0) = 0 \]

### Final Answer
Therefore, \(\frac{dy}{dx}\) at \(x = 1\) is:
\[ \boxed{0} \]