Question

Differentiate \[\tan \left( e^{\sin x }\right)\] ?

Answer 1

\[\text{Let y} = \tan\left( e^{\sin x} \right)\]

\[\text{Differentiate it with respect to x we get}, \]

\[ \frac{d y}{d x} = \frac{d}{dx}\left[ \tan\left( e^{\sin x} \right) \right]\]

\[ = \sec^2 \left( e^{\sin x} \right)\frac{d}{dx}\left( e^{\sin x } \right) \left[ \text{Using chain rule} \right]\]

\[ = \sec^2 \left( e^{\sin x} \right) \times e^{\sin x } \times \frac{d}{dx}\left( {\sin x} \right)\]

\[ = \cos x \sec^2 \left( e^{\sin x} \right) \times e^{\sin x} \]

\[So, \frac{d}{dx}\left\{ \tan\left( e^{\sin x} \right) \right\} = \cos x \sec^2 \left( e^{\sin x} \right) e^{\sin x}\]