`"If "y=sqrt(log{sin((x^(3))/(3)-1)}}," then find "(dy)/(dx).`

Explanation

`y=sqrt(log{sin((x^(3))/(3)-1)}}`
Putting `(x^(3))/(3)-1=v,` we get sin `((x^(2))/(3)-1)` = sin v = u.
Putting log `{sin((x^(2))/(3)-1)} = log y = z, we get y =sqrt(z),z = log u.`
`u=sin v, and v=(x^(2))/(3)-1.` Therefore,
`(dy)/(dx)(1)(2sqrt(z)), (dz)/(du) =(1)/(u), (du)/(dv) = cos v, and (dv)/(dx)=(2x)/(3)`
`Now,(dy)/(dx)=(dy)/(dz)xx(dz)/(du)xx(du)/(dv)xx(dv)/(dx)`
`=((1)/(2sqrt(z))((1)/(u))(cos v) ((2x)/(3))`
`=(x)/(3).(cos v)/(usqrt(log u))`
`=(x cot((x^(2))/(3)1))/(3sqrt(log{sin((x^(2))/(3)-1)}})`