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Question
Differentiate \[e^{\sin^{- 1} 2x}\] ?
Solution
\[\text{Let} y = e^{\sin^{- 1} 2x} \]
Differentiate it with respect to x we get,
\[\frac{d y}{d x} = \frac{d}{dx}\left( e^{\sin^{- 1} 2x} \right)\]
\[ = e^{\sin^{- 1} 2x} \times \frac{d}{dx}\left( \sin^{- 1} 2x \right) \left[ \text{Using chain rule} \right]\]
\[ = e^{\sin^{- 1} 2x} \times \frac{1}{\sqrt{1 - \left( 2x \right)^2}}\frac{d}{dx}\left( 2x \right)\]
\[ = \frac{2 e^{\sin^{- 1} 2x}}{\sqrt{1 - 4 x^2}}\]
\[So, \frac{d}{dx} \left( e^{\sin^{- 1} 2x} \right) =\frac{2 e^{\sin^{- 1} 2x}}{\sqrt{1 - 4 x^2}}\]
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