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

Answer

N/a

Explanation

`y = tan^(-1)((3a^(2)x-x^(3))/(a^(3)-3ax^(2)))`
Put `x = a tan theta rArr theta = tan^(-1)'x/a`
`:. y = tan^(-1)[(3tantheta-tan^(3)theta)/(1-3tan^(2)theta)], [:' tan3theta = (3tantheta-tan^(3)theta)/(1-3tan^(2)theta)]`
`=tan^(-1)(tan3theta) = 3theta`
`= 3tan^(-1)'(x)/(a), [:' theta = tan^(-1)'x/a]`
`:. (dy)/(dx) = 3.(d)/(dx) tan^(-1)'(x)/(a)=3.[(1)/(1+(x^(2))/(a^(2)))].(d)/(dx).(x/a)`
`= 3.(a^(2))/(a^(2)+x^(2)).(1)/(a) = (3a)/(a^(2)+x^(2))`