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Question
Find the derivatives of the following:
If y = `(sin^-1 x)/sqrt(1 - x^2)`, show that (1 – x2)y2 – 3xy1 – y = 0
Solution
y = `(sin^-1 x)/sqrt(1 - x^2)`
`y sqrt(1 - x^2) = sin^-1x`
Differentiating with respect to x
`y xx 1/2 (1 - x^2)^(1/2 - 1) (- 2x) + sqrt(1 x^2) y_1 = 1/sqrt(1 - x^2)`
`- xy(1 - x^2)^(- 1/2) + sqrt(1 - x^2) y_1 = 1/sqrt(1 - x^2)`
`(1 - x^2)^(1/2) [- xy(1 - x^2)^(- 1/2) + (1 - x^2)^(1/2) y_1]` = 1
`- xy (1 - x^2)^(- 1/2) xx (1 - x^2)^(1/2) + (1 - x^2)^(- 1/2) * (1 - x^2)^(1/2) y_1` = 1
`- xy + (1 - x^2) y_1` = 1
Differentiating with respect to x, we get
`- x * y_1 + y(-1) + (1 - x^2)y_2 + y_1 (0 - 2x)` = 0
`- xy_1 - y + (1 - x^2)y_2 - 2xy_1` = 0
`(1 - x^2)y_2 - 3xy_1 - y` = 0
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