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Question
Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?
Solution
\[\text{Let }y = \left( \sin^{- 1} x^4 \right)^4\]
Differentiate it with respect to x we get,
\[\frac{d y}{d x} = \frac{d}{dx} \left( \sin^{- 1} x^4 \right)^4 \]
\[ = 4 \left( \sin^{- 1} x^4 \right)^3 \frac{d}{dx}\left( \sin^{- 1} x^4 \right) \left[ \text{ Using chain rule} \right]\]
\[ = 4 \left( \sin^{- 1} x^4 \right)^3 \frac{1}{\sqrt{1 - \left( x^4 \right)^2}}\frac{d}{dx}\left( x^4 \right) \left[ \text{Using chain rule} \right]\]
\[ = 4 \left( \sin^{- 1} x^4 \right)^3 \frac{4 x^3}{\sqrt{1 - x^8}}\]
\[ = \frac{16 x^3 \left( \sin^{- 1} x^4 \right)^3}{\sqrt{1 - x^8}}\]
\[So, \frac{d}{dx} \left( \sin^{- 1} x^4 \right)^4 = \frac{16 x^3 \left( \sin^{- 1} x^4 \right)^3}{\sqrt{1 - x^8}}\]
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