tan inverse [ x ^1/3 + a^1/3 / 1 - x^1/3 a^1/3 ] - Maths - Continuity and Differentiability

Question

tan inverse [ x ^1/3 + a^1/3 / 1 - x^1/3 a^1/3 ]

Vijay Kumar Gupta · Accepted Answer

Consider the following expression
   y=tan-1x13-a131-x13a13Put x=tan3θ     and a=tan3ϕThis implies that,    x13=tanθ     and a13=tanϕThis further implies that,     θ=tan-1 x13     and    ϕ=tan-1a13So the given expression becomesy=  tan-1x13-a131-x13a13= tan-1tan3θ13-tan3ϕ131-tan3θ13tan3ϕ13                                        = tan-1tanθ-tanϕ1-tanθ tanϕ                                         = tan-1tanθ+ϕ                                       =θ+ϕ                                       =tan-1 x13+tan-1a13Differentiate both sides with respect to x
The second term being constant has derivative zero
     dydx=ddxtan-1 x13   +0             =11+x132  ddx  x13             =11+x23   13x-23            =131x231+x23            =131x23+x43

tan inverse [ x ^1/3 + a^1/3 / 1 - x^1/3 . a^1/3 ]