Differentiate the following with respect to x | vedclass

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(A) Let $y = \frac{e^{x}}{\sin x}$.Using the quotient rule $\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^{2}}$

Vedclass · Accepted Answer

$\frac{e^{x}(\sin x - \cos x)}{\sin^{2} x}$

Vedclass · Answer

(A) Let $y = \frac{e^{x}}{\sin x}$.Using the quotient rule $\frac{d}{dx} \left( \frac{u}{v} ight) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^{2}}$,where $u = e^{x}$ and $v = \sin x$:$\frac{dy}{dx} = \frac{\sin x \cdot \frac{d}{dx}(e^{x}) - e^{x} \cdot \frac{d}{dx}(\sin x)}{(\sin x)^{2}}$Since $\frac{d}{dx}(e^{x}) = e^{x}$ and $\frac{d}{dx}(\sin x) = \cos x$,we get:$\frac{dy}{dx} = \frac{\sin x \cdot e^{x} - e^{x} \cdot \cos x}{\sin^{2} x}$Factoring out $e^{x}$ from the numerator:$\frac{dy}{dx} = \frac{e^{x}(\sin x - \cos x)}{\sin^{2} x}$,where $x 
eq n\pi, n \in Z$.

Differentiate the following with respect to $x$: $\frac{e^{x}}{\sin x}$

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