The given equation is,
y= tan −1 ( 3x− x 3 1−3 x 2 )
Substitute x=tanθ.
y= tan −1 ( 3x− x 3 1−3 x 2 ) = tan −1 ( 3( tanθ )− tan 3 θ 1−3 tan 2 θ ) = tan −1 ( tan3θ ) =3θ
Substitute θ= tan −1 x in the above equation.
y=3 tan −1 x
Differentiate the above equation with respect to x.
y=3 tan −1 x dy dx =3 d dx ( tan −1 x ) = 3 1+ x 2
Thus, the derivative of y= tan −1 ( 3x− x 3 1−3 x 2 ) is dy dx = 3 1+ x 2 .