Mark the correct alternative in of the following:

If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at *x* = 1 is

#### Options

1

\[\frac{1}{2}\]

\[\frac{1}{\sqrt{2}}\]

0

#### Solution

\[y = \sqrt{x} + \frac{1}{\sqrt{x}}\]

\[ = x^\frac{1}{2} + x^{- \frac{1}{2}}\]

Differentiating both sides with respect to *x*, we get

\[\frac{dy}{dx} = \frac{d}{dx}\left( x^\frac{1}{2} + x^{- \frac{1}{2}} \right)\]

\[ = \frac{d}{dx}\left( x^\frac{1}{2} \right) + \frac{d}{dx}\left( x^{- \frac{1}{2}} \right)\]

\[ = \frac{1}{2} x^\frac{1}{2} - 1 + \left( - \frac{1}{2} \right) x^{- \frac{1}{2} - 1} \left( y = x^n \Rightarrow \frac{dy}{dx} = n x^{n - 1} \right)\]

\[ = \frac{1}{2} x^{- \frac{1}{2}} - \frac{1}{2} x^{- \frac{3}{2}}\]

Putting *x* = 1, we get

\[\left( \frac{dy}{dx} \right)_{x = 1} = \frac{1}{2} \times 1 - \frac{1}{2} \times 1 = 0\]

Thus, \[\frac{dy}{dx}\] 1 is 0.

Hence, the correct answer is option (d).