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Question
If y = sin–1x, then show that `(1 - x^2) (d^2y)/(dx^2) - x * dy/dx` = 0
Solution
y = sin–1x
Differentiating w.r.t. x,
`dy/dx = 1/sqrt(1 - x^2)`
∴ `sqrt(1 - x^2) * dy/dx` = 1
Squaring on both sides, we get
∴ `(1 - x^2) (dy/dx)^2 = 1`
Again differentiating w.r.t. x
`(1 - x^2)2 dy/dx * (d^2y)/(dx^2) + (dy/dx)^2 (-2x) = 0`
Dividing by `2 dy/dx`, we get
`(1 - x^2) (d^2y)/(dx^2) - x dy/dx = 0`
Hence proved.
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