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Question
Differentiate \[e^{\sin x }+ \left( \tan x \right)^x\] ?
Solution
\[\text{ Let y } = e^{\sin x} + \left( \tan x \right)^x \]
\[ \Rightarrow y = e^{ \sin x } + e^{\log \left( \tan x \right)^x } \]
\[ \Rightarrow y = e^{\sin x }+ e^{x\log\left( \tan x \right)} \]
Differentiating with respect to x
\[\frac{dy}{dx} = \frac{d}{dx}\left( e^{\sin x} \right) + \frac{d}{dx}\left\{ e^{x\log\left( \tan x \right)} \right\}\]
\[ = e^{ \sin x } \frac{d}{dx}\left( \sin x \right) + e^{x\log\left( \tan x \right)} \frac{d}{dx}\left( x \log\tan x \right) \]
\[ = e^{\sin x } \left( \cos x \right) + e^{\log \left( \tan x \right)^x }\left[ x\frac{d}{dx}\left( \log\tan x \right) + \log\tan x\frac{d}{dx}\left( x \right) \right]\]
\[ = e^{\sin x} \left( \cos x \right) + \left( \tan x \right)^x \left[ \frac{x}{\tan x}\left( \sec^2 x \right) + \log\tan x \right]\]
\[ = e^{\sin x } \left( \cos x \right) + \left( \tan x \right)^x \left[ x\sec x cosec x + \log\tan x \right]\]
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