Find the derivative with respect to x of the following:
(ii) `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 in exponent form
The function can be rewritten using exponents:
\[ y = x^{\frac{1}{2}} + x^{-\frac{1}{2}} \]

### Step 2: Differentiate the function
Now, we will differentiate \( y \) with respect to \( x \). We will use the power rule for differentiation, which states that if \( y = x^n \), then \( \frac{dy}{dx} = n \cdot x^{n-1} \).

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

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

### Step 3: Combine the derivatives
Now, we combine the results from the differentiation:
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{x}} - \frac{1}{2 x^{\frac{3}{2}}} \]

### Final Answer
Thus, the derivative of \( y = \sqrt{x} + \frac{1}{\sqrt{x}} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{x}} - \frac{1}{2 x^{\frac{3}{2}}} \]

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