1. ## differentiability

x( sqrt(x)-sqrt(x+1))
is this function is differentiable at x=0 (book answer is yes after rationalization)
my view
the domain does not allow negative values then how can we check differentiability at 0.

x( sqrt(x)-sqrt(x+1))
is this function is differentiable at x=0 (book answer is yes after rationalization)
my view
the domain does not allow negative values then how can we check differentiability at 0.
1. You determined the domain correctly but zero is a non-negative number.

2. Differentiate $f(x)=x( \sqrt{x}-\sqrt{x+1})$ wrt x and plug in x = 0. You should come out with $f'(0)=-1$

3. thanks sir
sir my simple question is that when we calculate LHD then we plug in 0-h ,h>0, h----0 (which is wrong) as negatives elements are not allowed.
according to you x^(3/2) is differentiable at 0

We don't. If a function is defined only for x>= a then the derivative at a is given by $\lim_{h\to a^+} \frac{f(a+ h)- f(a)}{h}$