For this function to be defined, we need that and .Thus, this function is defined on .
The function is a rational function, thus it is differentiable on its domain; the composition of this function with the square root function is differentiable where the rational function does not vanish. Finally is the product of this composition with a linear function, hence is differentiable on . Let us compute the first derivative:
Let us compute the limits of at the open endpoints of its domain:
We shall now look for oblique asymptotes:
We can now draw the variation table of (where and ):
Let us look for a point of inflexion. By arguments similar to those we used for differentiating , we prove that is differentiable on and we have:
Noah Dana-Picard 2007-12-28