[tex]DIFFERENTIATION \\ \\ \\ Let \: u \: = \: {e}^{ \sin(x) } \: \: and \: v = \cos(x) \\ \\ Then \: , \: \frac{du}{dx} \: = \: \cos(x) . {e}^{ \sin(x) } \\ \\ and \: \: \: \: \: \: \frac{dv}{dx} \: \: = \: - \sin(x) \\ \\ \\ Therefore \: , \\ \\ \\ \frac{du}{dv} \: = \: \frac{ \frac{du}{dx} }{ \frac{dv}{dx} } \: = \: \frac{ \cos(x) . {e}^{ \sin(x) } }{ - \sin(x) } \\ \\ \\ \frac{du}{dv} \: = \: - \cot(x) . {e}^{ \sin(x) } \: \: \: \: \: \: \: \: \: \: Ans.[/tex]
