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If y = (sin–1 x)2, prove that: (1–x2) y2–xy1–2=0

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 Note: y2 represents second order derivative i.e.\(\frac{d^2y}{dx^2}\) and y1 = dy/dx

Given,

y = (sin–1 x)2 ……equation 1

to prove : (1–x2) y2–xy1–2=0

We notice a second–order derivative in the expression to be proved so first take the step to find the second order derivative.

Let’s find \(\frac{d^2y}{dx^2}\)

Using chain rule we will differentiate the above expression

Let t = sin–1 x =>\(\frac{dt}{dx}=\frac{1}{\sqrt{1-x^2}}\) [using formula for derivative of sin–1x]

And y = t2

 Again differentiating with respect to x applying product rule:

∴ (1–x2) y2–xy1–2=0 ……proved

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