Differentiate `tan^(-1)((3a^2x-x^3)/(a^3-3a x^2))`,` -1/(sqrt(3))ltx/alt1/(sqrt(3))`

Explanation

Given that,
`y=tan^(-1)((3a^2x-x^3)/(a^3-3a x^2))`,
` -1/(sqrt(3))ltx/alt1/(sqrt(3))`
`=>y=tan^-1[(3(x/a)-x^3/a^3)/(1-3x^2/a^2)]`
`x/a=tan theta`
`y=tan^-1[(3tantheta-tan^3 theta)/(1-3tan^2 theta)]`
`=>y=tan^-1(tan 3 theta)`
` -1/(sqrt(3))ltx/alt1/(sqrt(3))`
` =>-1/(sqrt(3))lt tan theta lt1/(sqrt(3))`
`=>-pi/6 lt theta lt pi/6`
`=>-pi/2 lt 3theta lt pi/2`
`y=3 theta `
`=>y= 3 tan^-1(x/a)`
`(dy)/(dx)=3*d/(dx)[ tan^-1 (x/a)]`
`=3/(1+(x/a)^2) * d/(dx) (x/a)`
`=(3a)/(x^2+a^2)`