Question

Differentiate \[e^{\sin} \sqrt{x}\] ?

Answer 1

\[Let y = e^{\sin}\sqrt{x} \]

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

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

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

\[ = e^{\sin}\sqrt{x} \times \cos\sqrt{x}\frac{d}{dx}\sqrt{x} \left[ \text{using chain rule } \right]\]

\[ = e^{\sin }\sqrt{x} \times \cos\sqrt{x} \times \frac{1}{2\sqrt{x}}\]

\[ = \frac{\cos\sqrt{x} e^{\sin}\sqrt{x}}{2\sqrt{x}}\]

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