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Question
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
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Solution
\[\frac{d}{dx}\left[ \left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right) \right]\]
\[ = \frac{d}{dx}\left[ \left( x + x^{- 1} \right)\left( x^\frac{1}{2} + x^\frac{- 1}{2} \right) \right]\]
\[ = \frac{d}{dx}\left( x^\frac{3}{2} + x^\frac{1}{2} + x^\frac{- 1}{2} + x^\frac{- 3}{2} \right)\]
\[ = \frac{d}{dx}\left( x^\frac{3}{2} \right) + \frac{d}{dx}\left( x^\frac{1}{2} \right) + \frac{d}{dx}\left( x^\frac{- 1}{2} \right) + \frac{d}{dx}\left( x^\frac{- 3}{2} \right)\]
\[ = \frac{3}{2} x^\frac{1}{2} + \frac{1}{2} x^\frac{- 1}{2} - \frac{1}{2} x^\frac{- 3}{2} - \frac{3}{2} x^\frac{- 5}{2} \]
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