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Question
Differentiate the following w.r.t. x :
`cos^-1(sqrt(1 - cos(x^2))/2)`
Solution
Let y = `cos^-1(sqrt(1 - cos(x^2))/2)`
= `cos^-1(sqrt((2sin^2(x^2/2))/2))`
= `cos^-1[sin(x^2/2)]`
= `cos^-1[cos(pi/2 - x^2/2)]`
= `pi/(2) - x^2/(2)`
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"(pi/2 - x^2/2)`
= `"d"/"dx"(pi/2) - (1)/(2)"d"/"dx"(x^2)`
= `0 - (1)/(2) xx 2x`
= – x.
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