I need some clarifications about the correct way to solve an exercise.

I have this function:
$$f(x,y)=\frac{(x-1)y^2}{\sin^2\sqrt{(x-1)^2+y^2}}$$

and I have to 

1. analyse the existence of partial derivatives and the differentiability in (1,0)
2. say if f is differentiable in (0,0)

**About point 1**:
I have this doubt. Do I have to calculate the partial derivatives considering the expression of g(x,y) in (1,0) (i.e. g(1,0)=0 -> $\displaystyle {\partial f}/\partial x=\partial 0 / \partial x=0={\partial f}/\partial y $ (\*) or do I have to use 

$\displaystyle\frac{\partial}{\partial x}g(1,0)=\lim\_{t->0} \frac{f(1+t,0)-g(1,0)}{t}$ ? (\*\*) Or do I have to find the expressions of the partial derivatives without using the limit and *then* do the limit for (x,y)->(1,0)?

If I want to analyse if $\frac{\partial}{\partial x}g(1,0)$ is continue, do I have to compare (\*) and (\*\*)?

**About point 2**:
If a function has continue partial derivatives in a point, it is differentiable in that point.

So I have thought that in Dom f, the partial derivatives are continue because thay're composition of continue function. Since $0 \in Dom f$, g(x,y) is differentiable in (0,0). 

Is this reasoning ok, or do I have to check with calculation?

Many thanks for your help!