[" (c) "pi+3" tan "x],[" 53.If "x>-(1)/(sqrt(3))," then "tan^(-1)((3x-x^(3))/(1-3x^(2)))" equals "],[[" (a) "3tan^(-1)x," (b) "-pi+3tan^(-1)x]]

If x lt -(1)/sqrt(3) , then tan^(-1)(3x-x^(3))/(1-3x^(2)) equals

If x gt (1)/sqrt(3) then tan^(-1) (3x-x^(3))/(1-3x^(2)) equals

If -(1)/sqrt(3) lt x lt (1)/sqrt(3), then tan^(-1) (3x-x^(3))/(1-3x^(2)) equals

tan^(-1)((sqrt(x)(3-x))/(1-3x))

Prove that 3tan^(-1)x=tan^(-1)((3x-x^3)/(1-3x^2))

Differentiate tan^(-1)((3x-x^(3))/(1-3x^(2))), if x>(1)/(sqrt(3))

Differentiate tan^(-1)((3x-x^(3))/(1-3x^(2))), if x<-(1)/(sqrt(3))

y = tan^(-1)((3x-x^3)/(1-3x^2)), 1/(sqrt3) , x , 1/(sqrt3) .

Prove that 3 tan^(-1) x= {(tan^(-1) ((3x - x^(3))/(1 - 3x^(2))),"if " -(1)/(sqrt3) lt x lt (1)/(sqrt3)),(pi + tan^(-1) ((3x - x^(3))/(1 - 3x^(2))),"if " x gt (1)/(sqrt3)),(-pi + tan^(-1) ((3x - x^(3))/(1 - 3x^(2))),"if " x lt - (1)/(sqrt3)):}