`y=tan^(-1)` `((3x-x^3)/(1-3x^2))`. Find `dy/dx` .

Options

A. `(3)/(1+9x^(2))" for all "x in R`
B. `(3)/(1+x^(2))" for all "x in R`
C. `(3)/(1+x^(2))" for all "x in R-{-(1)/(sqrt(3)),(1)/(sqrt(3))}`
D. `(1)/(9+x^(2))`

Answer

C

Explanation

We have,
`f(x)=tan^(-1)((3x-x^(3))/(1-3x^(2)))`
`implies" "f(x)={{:(3tan^(-1)x+pi," for "xlt-(1)/(sqrt(3))),(3tan^(-1)x," for "-(1)/(sqrt(3))ltxlt(1)/(sqrt(3))),(3tan^(-1)x-pi," for "xgt(1)/(sqrt(3))):}`
`:." "f'(x)=(3)/(1+x^(2))" for all "x inR-{-(1)/(sqrt(3)),(1)/(sqrt(3))}`