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Question
Find the derivatives of the following:
If y = `(cos^-1 x)^2`, prove that `(1 - x^2) ("d"^2y)/("d"x)^2 - x ("d"y)/("d"x) - 2` = 0. Hence find y2 when x = 0
Solution
y = `(cos^-1 x)^2`
`("d"y)/("d"x) = 2(cos^-1_x) xx 1/(- sqrt(1 - x^2)`
`sqrt(1 - x^2) ("d"y)/("d"x) = - 2 cos^-1 x`
`sqrt(1 - x^2) * ("d"^2y)/("d"x)^2 + ("d"y)/("d"x) xx 1/2 (1 - x^2)^(1/2 - 1) (- 2x) = - 2xx 1/sqrt(1 - x^2)`
`sqrt(1 - x^2) ("d"^2y)/("d"x^2) - x (1 - x^2)^(1/2) ("d"y)/("d"x) = 2/sqrt(1 - x^2)`
`sqrt(1 - x^2) ("d"^2y)/("d"x^2) - x/sqrt(1 - x^2) ("d"y)/("d"x) = 2/sqrt(1 - x^2)`
`sqrt(1 - x^2) [sqrt(1 - x^2) ("d"^2y)/("d"x) - x/sqrt(1 - x^2) ("d"y)/("d"x)]` = 2
`(1 - x^2) ("d"^2y)/("d"x^2) - x * ("d"y)/("d"x)` = 2
`(1 - x^2) ("d"^2y)/("d"x^2) - x ("d"y)/("d"x) - 2` = 0
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