Question

If$\sf{y={\sin}^{-1}\{x-\sqrt{1-x}-\sqrt{x}\sqrt{1-x^2}\}}$ and 0 < x < 1, then find$\sf{\frac{dy}{dx}}$.
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Answer 1

Find derivative of f(x)=sin−1[x1−x−−−−−√−x−−√1−x2−−−−−√], 0<x<1

Let x=sina and x−−√=cosb

Then I'll get:

y=sin−1[sinacosb−cosasinb]=sin−1[sin(a−b)]⟹siny=sin(a−b)⟹y=nπ+(−1)n(a−b)=nπ+(−1)n(sin−1x−sin−1x−−√)

Thus,

y′=ddx[nπ+(−1)n(a−b)]=⎧⎩⎨⎪⎪⎪⎪11−x2√−12x√1−x√ if n is even−[11−x2√−12x√1−x√] if n is odd

Answer 2

Answer:

tex]\sf{y={\sin}^{-1}\{x-\sqrt{1-x}-\sqrt{x}\sqrt{1-x^2}\}}[/tex] and 0 < x < 1, then find$\sf{\frac{dy}{dx}}$.

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$\pink\dashrightarrow\bf{Full\:Explanation\:Required}$