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Question
Differentiate the following w.r.t.x:
`log(sqrt((1 - cos3x)/(1 + cos3x)))`
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Solution
Let y = `log(sqrt((1 - cos3x)/(1 + cos3x)))`
= `log(sqrt((2sin^2((3x)/2))/(2cos^2((3x)/2))))`
= `logtan((3x)/2)`
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"[logtan((3x)/2)]`
= `(1)/(tan((3x)/2)) xx "d"/"dx"[tan((3x)/2)]`
= `(1)/(tan((3x)/2)) xx sec^2((3x)/2)."d"/"dx"((3x)/2)`
= `cos((3x)/2)/(sin((3x)/2)) xx (1)/(cos^2((3x)/2)) xx (3)/(2) xx 1`
= `3 xx (1)/(2sin((3x)/2)cos((3x)/2)`
= `3 xx (1)/(sin3x)`
= 3 cosec 3x
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