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The derivative of $x^{2} a^{x} \log x \tan x$ with respective to $x$ is

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$x^{2} a^{x} \log x \tan x [\frac{2}{x} - \log a + \frac{1}{x \log x} + 2 cosec 2 x]$

$x^{2} a^{x} \log x \tan x [\frac{2}{x} + \log a + \frac{1}{x \log x} - 2 cosec 2 x]$

$x^{2} a^{x} \log x \tan x [\frac{2}{x} + \log a + \frac{1}{x \log x} + 2 cosec 2 x]$

$x^{2} a^{x} \log x \tan x [\frac{2}{x} + \log a - \frac{1}{x \log x} - 2 cosec 2 x]$

answer is D.

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Detailed Solution

$y = x^{2} a^{x} \log x \tan x$

$y' = x^{2} a^{x} . \log x . \sec^{2} x + x^{2} . a^{x} . \frac{1}{x} . \tan x + x^{2} . a^{x} \log a . \log x . \tan x + 2 x . a^{x} . \log x . \tan x$

$= x^{2} a^{x} \log x . \tan x [\frac{1}{\sin x . \cos x} + \frac{1}{x \log x} + \log a + \frac{2}{x}]$

$= x^{2} a^{x} \log x . \tan x [2 cosec 2 x + \frac{1}{x \log x} + \log a + \frac{2}{x}]$

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