The function $f(x)=x|x|$ is is differentiable everywhere. For $x>0$, you have $f(x)=x^2$, differentiable. For $x<0$ you have $f(x)=-x^2$, differentiable. At $0$, you have
so the derivative exists and is zero.
For two functions, using the Wronskian is overkill. Linear dependence for two functions means that one is a multiple of the other: this is trivial to check for yes or for no. In your example, for instance, if $x|x|=cx^2$ for all $x$, then evaluate at $1$ to get $c=1$, and at $-1$ to get $c=-1$; so such $c$ cannot exist and the functions are linearly independent.