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Question

If y=cos1x, find d2ydx2 in terms of y alone.

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Solution

y=cos1x
dydx=11x2=(1x2)12
d2ydx2=ddx[(1x2)12]
=(12)(1x2)32.ddx(1x2)
= 121x2)3×(2x)
d2ydx2=x2(1x2)3 .....(i)
Now y=cos1xx=cosy
Putting x=cosy in equation (i), we obtain
d2ydx2=cosy(1cos2y)3
=cosysin3y
=cosysiny×1sin2y
d2ydx2=coty.cosec2y

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