Question

Differentiate the following w.r.t.x:

`log(sqrt((1 - cos3x)/(1 + cos3x)))`

Answer 1

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