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y = sin ^(-1)(2xsqrt(1 - x^(2))),-(1)/sq...

`y = sin ^(-1)(2xsqrt(1 - x^(2))),-(1)/sqrt(2) lt x lt (1)/sqrt(2)`

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To solve the problem step by step, we will differentiate the function \( y = \sin^{-1}(2x\sqrt{1 - x^2}) \) for \( -\frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}} \). ### Step 1: Substitute \( x \) with \( \sin \theta \) Let \( x = \sin \theta \). Then, we can express \( y \) in terms of \( \theta \): \[ y = \sin^{-1}(2\sin \theta \sqrt{1 - \sin^2 \theta}) ...
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