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Question
Differentiate \[e^{ax} \sec x \tan 2x\] ?
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Solution
\[\text{Let } y = e^{ax} \sec x \tan2x\]
Differentiate it with respect to x,
\[\frac{d y}{d x} = \frac{d}{dx}\left( e^{ax} sec x \tan2x \right)\]
\[ = e^{ax} \frac{d}{dx}\left\{ \sec x \tan2x \right\} + \sec x \tan2x\frac{d}{dx}\left\{ e^{ax} \right\} \]
\[ = e^{ax} \left[ \sec x \tan x \tan2x + 2 \sec^2 2x\sec x \right] + a e^{ax} \sec x \tan2x \]
\[ = e^{ax} \left[ \sec x \tan x \tan2x + 2 \sec^2 2x\sec x \right] + a e^{ax} sec x \tan2x\]
\[ = a e^{ax} \sec x \tan2x + e^{ax} \sec x \tan x \tan2x + 2 e^{ax} \sec x \sec^2 2x\]
\[ = e^{ax} \sec x\left\{ a \tan2x + \tan x \tan2x + 2 \sec^2 2x \right\}\]
\[So, \frac{d}{dx}\left( e^{ax} \sec x \tan2x \right) = e^{ax} \sec x\left\{ a \tan2x + \tan x \tan2x + 2 \sec^2 2x \right\}\]
