Show function is differentiable over its domain

I need to show that:

is differentiable over its domain. The only point I see there being a problem at is when (x,y,z) = (0,0,0).

Then is it sufficient to show that:

where for this case.

I do see a problem in the h(x,y,z) function regarding f(0,0,0). That's what I am trying to find the limit of, and in my function I'd get 0/0 when I plug in those values. Also, I am getting the partial derivatives to be of the form:

Using all the info I seem to get many terms who wind up becoming 0/0. I believe I am missing the point and/or missing a way to simplify everything. Thanks for any help.