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Question

If y=(sin1x)2, then prove that (1x2)d2ydx2xdydx2=0

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Solution

Given y=(sin1x)2

On differentiating y with respect to x.

dydx=2sinx1x2

On again differentiating dydx with respect to x.

2yx2=2[1x2.11x2+sin1x(1x2)12.122x](1x2)2

2[1+x.sin1x1x2]1x2=21x2+2xsin1x1x2(1x2)

On putting value of dyd and 2yx2 in LHS

(1x2)2yx2xdydx2=0

=(1x2)21x2+2xsin1x1x2(1x2)x2xsin1x1x22

=21x21x2+2xsin1x1x22xsin1x1x22

=0 [henceproved]


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