Question

differentiate sinx with respect to e^x​

Answer 1

Answer:

See The Attachment My Friend.......

Answer 2

Answer :-

$\implies \boxed{\sf{ \frac{dy}{dt} = \frac{ \cos x}{ {e}^{x}}}} \\ \bf{ or} \\ \implies \boxed{\sf{ \frac{d (\sin x)}{d ( {e}^{x} )} = \frac{ \cos x}{ {e}^{x} } }} \:$

Explanation :-

$\implies \sf{let \: y \: = \sin x} \\ \\ \sf{differentiate \: with \: respect \: to \: x} \\ \\ \implies \sf{ \frac{dy}{dx} = \frac{d}{dx} \cos x \: } \: \\ \boxed{ \sf{\: \because \: \frac{d}{d x} \sin x = \cos x}} \\ \\ \implies \sf{ \frac{dy}{dx} = \cos x \: \: \:........ \: eq.(1)}$

again let ,

$\to \sf{ t \: = {e}^{x} } \\ \\ \sf{ differentiate \: with \: respect \: to \: x} \\ \\ \implies \sf{ \frac{dt}{dx} = \frac{d}{dx} {e}^{x} } \\ \\ \implies \sf{ \frac{dt}{dx} = {e}^{x} \: \: .....eq.(2)} \\ \\ \sf{ apply \to \frac{eq.(1)}{eq.(2)} } \\ \\ \implies \sf{ \frac{ \frac{dy}{dx} }{ \frac{dt}{dx} } = \frac{ \cos x}{ {e}^{x} } } \\ \\ \implies \boxed{\sf{ \frac{dy}{dt} = \frac{ \cos x}{ {e}^{x}}}} \\ \bf{ or} \\ \implies \boxed{\sf{ \frac{d (\sin x)}{d ( {e}^{x} )} = \frac{ \cos x}{ {e}^{x} } }}$