Determine at what points in its domain $f$ is differentiable

Question

I have been trying to solve the two problems. Here's what I did:

1) I found the partial derivatives:

(A) $\,f_x = -y\sin(xy)$ and $f_y = -x\sin(xy)$

(B) $\,f_x = ye^{xy} + 2x/(1+x^2+y^2)$ and $f_y = xe^{xy} + 2y/(1+x^2+y^2)$

2) I think that there is a theorem that says that if the partial derivatives of a function are continuous at $(a,b)$ then the function is differentiable at (a,b).

Now I don't know how to show that partial derivatives are continuous.

Is it correct for me to assume that since all the functions are continuous, so their product is also continuous, and hence both partial derivatives are continuous over the domain so the function is differentiable over the domain?

Answer 1

Since,

both partial derivatives exist and are continuous for both part(A) and part (B) in their whole domain $R^2$, so both functions are differentiable at every point in $R^2$.