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Question
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
Solution
\[ = \frac{d}{dx}\left[ log \left( x^\frac{- 1}{2} \right) \right] + 5\frac{d}{dx}\left( x^a \right) - 3\frac{d}{dx}\left( a^x \right) + \frac{d}{dx}\left( x^\frac{2}{3} \right) + 6\frac{d}{dx}\left( x^\frac{- 3}{4} \right)\]
\[ = \frac{d}{dx}\left( \frac{- 1}{2}\log x \right) + 5\frac{d}{dx}\left( x^a \right) - 3\frac{d}{dx}\left( a^x \right) + \frac{d}{dx}\left( x^\frac{2}{3} \right) + 6\frac{d}{dx}\left( x^\frac{- 3}{4} \right)\]
\[ = \frac{- 1}{2} . \frac{1}{x} + 5a x^{a - 1} - 3 a^x \log a + \frac{2}{3} x^\frac{- 1}{3} + 6\left( \frac{- 3}{4} \right) x^\frac{- 7}{4} \]
\[ = \frac{- 1}{2x} + 5a x^{a - 1} - 3 a^x \log a + \frac{2}{3} x^\frac{- 1}{3} - \frac{9}{2} x^\frac{- 7}{4} \]
\[\]
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