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Question
Differentiate \[\tan \left( e^{\sin x }\right)\] ?
Solution
\[\text{Let y} = \tan\left( e^{\sin x} \right)\]
\[\text{Differentiate it with respect to x we get}, \]
\[ \frac{d y}{d x} = \frac{d}{dx}\left[ \tan\left( e^{\sin x} \right) \right]\]
\[ = \sec^2 \left( e^{\sin x} \right)\frac{d}{dx}\left( e^{\sin x } \right) \left[ \text{Using chain rule} \right]\]
\[ = \sec^2 \left( e^{\sin x} \right) \times e^{\sin x } \times \frac{d}{dx}\left( {\sin x} \right)\]
\[ = \cos x \sec^2 \left( e^{\sin x} \right) \times e^{\sin x} \]
\[So, \frac{d}{dx}\left\{ \tan\left( e^{\sin x} \right) \right\} = \cos x \sec^2 \left( e^{\sin x} \right) e^{\sin x}\]
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