Derivatives of ` y = cos^(-1) sqrt((cos 3x)/(cos ^(3) x))` with respect to x , is

Options

A. `sqrt((6)/(cos 4x + cos 2x))`
B. `(6)/sqrt(cos 4x + cos 2x)`
C. `(sqrt6)/sqrt(cos 4x + cos 2x)`
D. None of these

Answer

A

Explanation

`y = cos^(-1) sqrt((cos 3x)/(cos^(3) x)) rArr cos y = sqrt((cos 3x)/(cos^(3) x))`
` rArr cos ^(2) y = (4 cos ^(3) x - 3 cos x)/(cos^(3) x )`
`= 4- 3 sec^(2) x = 1 - 3 tan^(2) x ` ...(ii)
` rArr sin^(2) y = 3 tan^(2) x rArr sin y = sqrt(3) tan x `
On differentiating both sides w.r.t.x we get
`cos y.(dy)/(dx) = sqrt(3). sec^(2) x `
` rArr (dy)/(dx) = (sqrt(3) sec^(2) x )/(cos y )=sqrt((3)/(cos^(2) ycos ^(4) x))`
`sqrt((3)/((1 - 3 tan^(2) x ) cos^(4)x )) " " ` [ from Eq. (i) ]
`(dy)/(dx)= sqrt((3)/(cos^(2) xx(cos^(2)x- 3 sin^(2) x)))`
`sqrt((3)/(cos^(2) xx(4 cos^(2) x - 3)))`
`sqrt((3)/(cos xx(4 cos^(3) x - 3cos x)))`
` (dy)/(dx) = sqrt((3)/(cos x cos 3x))= sqrt((6)/(cos 4x + cos 2x))`