Differentiate the following w.r.t. x:


Differentiate the following w.r.t. x:

$\frac{e^{x}}{\sin x}$


Let $y=\frac{e^{x}}{\sin x}$

By using the quotient rule, we obtain

$\frac{d y}{d x}=\frac{\sin x \frac{d}{d x}\left(e^{x}\right)-e^{x} \frac{d}{d x}(\sin x)}{\sin ^{2} x}$

$=\frac{\sin x \cdot\left(e^{x}\right)-e^{x} \cdot(\cos x)}{\sin ^{2} x}$

$=\frac{e^{x}(\sin x-\cos x)}{\sin ^{2} x}, x \neq n \pi, n \in \mathbf{Z}$


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