Find, the derivative of the following w.r.t. x :
`sqrt(x)+1/sqrt(x)`

Answer

To find the derivative of the function \( y = \sqrt{x} + \frac{1}{\sqrt{x}} \) with respect to \( x \), we will follow these steps:

### Step 1: Rewrite the function
We start by rewriting the function in terms of exponents:
\[ y = x^{1/2} + x^{-1/2} \]

### Step 2: Differentiate using the power rule
Next, we differentiate each term using the power rule, which states that if \( y = x^n \), then \( \frac{dy}{dx} = n \cdot x^{n-1} \).

1. Differentiate \( x^{1/2} \):
\[ \frac{d}{dx}(x^{1/2}) = \frac{1}{2} x^{-1/2} \]

2. Differentiate \( x^{-1/2} \):
\[ \frac{d}{dx}(x^{-1/2}) = -\frac{1}{2} x^{-3/2} \]

### Step 3: Combine the derivatives
Now we combine the derivatives of both terms:
\[ \frac{dy}{dx} = \frac{1}{2} x^{-1/2} - \frac{1}{2} x^{-3/2} \]

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

### Step 5: Rewrite in terms of square roots
To express the derivative in terms of square roots, we can rewrite it as:
\[ \frac{dy}{dx} = \frac{1}{2} \left( \frac{1}{\sqrt{x}} - \frac{1}{x\sqrt{x}} \right) \]

### Final Answer
Thus, the derivative of the function \( y = \sqrt{x} + \frac{1}{\sqrt{x}} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{2} \left( \frac{1}{\sqrt{x}} - \frac{1}{x\sqrt{x}} \right) \]

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