Differentiate the following w.r.t.x: e^x - Vidyadip

Question

Differentiate the following w.r.t.x: $\frac{\text{e}^\text{x}}{\sin\text{x}}$

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Answer

$\text{Let}\ \text{y}=\frac{\text{e}^\text{x}}{\sin\text{x}}$
$\therefore\frac{\text{dy}}{\text{dx}}=\frac{\sin\text{x}.\frac{\text{d}}{\text{dx}}(\text{e}^{\text{x}})-\text{e}^{\text{x}}.\frac{\text{d}}{\text{dx}}(\sin\text{x})}{\sin^{2}\text{x}}$
$=\frac{\sin\text{x}.\text{e}^{\text{x}}-\text{e}^{\text{x}}.\cos\text{x}}{\sin^{2}\text{x}}=\text{e}^{\text{x}}\bigg(\frac{\sin\text{x}-\cos\text{x}}{\sin^{2}\text{x}}\bigg)$

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