If $$y = \frac{1}{{1 + {x^{\beta - \alpha }} + {x^{\gamma - \alpha }}}} + \frac{1}{{1 + {x^{\alpha - \beta }} + {x^{\gamma - \beta }}}} + \frac{1}{{1 + {x^{\alpha - \gamma }} + {x^{\beta - \gamma }}}}$$            then $$\frac{{dy}}{{dx}}$$  is equal to :

Solution :
$$\eqalign{ & {\text{We have,}} \cr & y = \frac{1}{{1 + \frac{{{x^\beta }}}{{{x^\alpha }}} + \frac{{{x^\gamma }}}{{{x^\alpha }}}}} + \frac{1}{{1 + \frac{{{x^\alpha }}}{{{x^\beta }}} + \frac{{{x^\gamma }}}{{{x^\beta }}}}} + \frac{1}{{1 + \frac{{{x^\alpha }}}{{{x^\gamma }}} + \frac{{{x^\beta }}}{{{x^\gamma }}}}} \cr & = \frac{{{x^\alpha }}}{{{x^\alpha } + {x^\beta } + {x^\gamma }}} + \frac{{{x^\beta }}}{{{x^\alpha } + {x^\beta } + {x^\gamma }}} + \frac{{{x^\gamma }}}{{{x^\alpha } + {x^\beta } + {x^\gamma }}} \cr & = \frac{{{x^\alpha } + {x^\beta } + {x^\gamma }}}{{{x^\alpha } + {x^\beta } + {x^\gamma }}} \cr & = 1 \cr & \therefore \,\frac{{dy}}{{dx}} = 0 \cr} $$