# Partial derivatives and Differentiability

I understand the gist of this question, however I don't understand how to conclude that say $f_1$ (the rest are similar) is differentiable at all points. I don't know which of the theorems the solution is talking about and I can't use the fact that the composite of differentiable functions is differentiable as $p_x$ is only continuous not differentiable.

First remark: $p_x$ is differentiable. Second remark: I am pretty sure that you studied a theorem saying that a differentiable functions always possesses partial derivative. This is the result the hint refers to.