Question
Differentiate the following w.r.t.x: $\text{e}^{\sin^{-1}\text{x}}$
✓
Answer
$\text{Let}\ \text{y}=\text{e}^{\sin^{-1}\text{x}}$
$\therefore\ \frac{\text{dy}}{\text{dx}}=\text{e}^{\sin^{-1}\text{x}}.\frac{\text{d}}{\text{dx}}{\sin^{-1}\text{x}}=\text{e}^{\sin^{-1}\text{x}}.\frac{1}{\sqrt{1-\text{x}^{2}}}\ \ \bigg[\because\frac{\text{d}}{\text{dx}}\text{e}^\text{f(x)}=\text{e}^\text{f(x)}\frac{\text{d}}{\text{dx}}\text{f(x)}\bigg]$
$\therefore\ \frac{\text{dy}}{\text{dx}}=\text{e}^{\sin^{-1}\text{x}}.\frac{\text{d}}{\text{dx}}{\sin^{-1}\text{x}}=\text{e}^{\sin^{-1}\text{x}}.\frac{1}{\sqrt{1-\text{x}^{2}}}\ \ \bigg[\because\frac{\text{d}}{\text{dx}}\text{e}^\text{f(x)}=\text{e}^\text{f(x)}\frac{\text{d}}{\text{dx}}\text{f(x)}\bigg]$
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