Question

Answer the above THE BEST WILL GET BRAINLIEST
Note: if you randomly wrote anything you will be reported
PS: You will not get the answer in google so don't waste your time!!​

Answer 1

GIVEN EXPRESSION :-

$\sf{\dfrac{d}{dx}\bigg(\sqrt{x}+\dfrac{1}{\sqrt{x} } \bigg)^2 }$

OBJECTIVE :-

To differentiate the given expression.

SOLUTION :-

$\sf{\dfrac{d}{dx}\bigg(\sqrt{x}+\dfrac{1}{\sqrt{x} } \bigg)^2 }$

Applying power rule :-

$\displaystyle{\sf{= 2\bigg(\sqrt{x}+\frac{1}{\sqrt{x} } \bigg).\frac{d}{dx}\bigg(\sqrt{x}+\frac{1}{\sqrt{x} }\bigg) }}$

$\displaystyle{\sf{= 2\bigg(\sqrt{x}+\frac{1}{\sqrt{x} } \bigg)\bigg(\frac{d}{dx}(\sqrt{x} ) +\frac{d}{dx}(\frac{1}{\sqrt{x} } ) \bigg)}}$

$\displaystyle{\sf{= 2\bigg(\sqrt{x} + \frac{1}{\sqrt{x} } \bigg)\bigg[\frac{1}{2}x^{\frac{1}{2}-1 }+\big(\frac{-1}{2} \big)x^{\frac{-1}{2}-1 } \bigg]}}$

$\displaystyle{\sf{= 2\bigg(\frac{1}{2\sqrt{x} }-\frac{1}{2x^\frac{3}{2} } \bigg)\bigg( \sqrt{x} + \frac{1}{\sqrt{x} } \bigg) }}$

On simplifying by taking LCM :-

$\displaystyle{\sf{= \frac{x^2-1}{x^2} }}$

Solving it :-

$\displaystyle{\sf{\implies \frac{x^2-1}{x^2} }=\frac{x^2}{x^2}-\frac{1}{x^2} }\\\\\boxed{\displaystyle{\sf{= 1-\frac{1}{x^2} }}}$

So, the answer is option (A).