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Question
If y = cos–1 x, find `(d^2y)/dx^2` in terms of y alone.
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Solution
Given, y = cos–1 x
⇒ x = cos y
Differentiating both sides with respect to x,
`d/dx (x) = d/dx cos y`
`1 = - sin y dy/dx`
`dy/dx = - 1/sin y`
`dy/dx` = −cosec y
Differentiating both sides again with respect to x,
`(d^2 y)/dx^2 = - d/dx` cosec y
= −(−cosec y cot y) `dy/dx`
= [cosec y cot y] (−cosec y) ...`["Substituting the value of" dy/dx]`
= −cosec2 y cot y
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