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