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Question
\[e^x \log \sqrt{x} \tan x\]
Solution
\[\text{ Let } u = e^x ; v = \log \sqrt{x}; w = \tan x\]
\[\text{ Then } , u' = e^x ; v' = \frac{1}{\sqrt{x}} \times \frac{1}{2\sqrt{x}} = \frac{1}{2x}; w' = \sec^2 x\]
\[\text{ Using the product rule }:\]
\[\frac{d}{dx}\left( uvw \right) = u'vw + uv'w + uvw'\]
\[ = e^x \log \sqrt{x}\tan x + e^x \times \frac{1}{2x}\tan x + e^x \log \sqrt{x} \sec^2 x\]
\[ = e^x \left( \log x^\frac{1}{2} . \tan x + \frac{\tan x}{2x} + \log x^\frac{1}{2} . \sec^2 x \right)\]
\[ = e^x \left( \frac{1}{2} \log x . \tan x + \frac{\tan x}{2x} + \frac{1}{2} \log x . \sec^2 x \right)\]
\[ = \frac{e^x}{2}\left( \log x . \tan x + \frac{\tan x}{x} + \log x . \sec^2 x \right)\]
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