How does one calculate the time derivative of
$$ x_{k+1} = C_1\, \text{sign}(x_k-y_k)\sqrt{2\vert x_k-y_k\vert}, $$
with respect to $x_k$ ?
I know that the right part of the equation should yield $$ \frac{\partial}{\partial\,x_k} \sqrt{2\vert x_k-y_k\vert} = \frac{x_k-y_k}{\vert x_k -y_k\vert^{\frac{3}{2}}}. $$
Yet, I don't know to get the final result. How can one differentiate the sign function?
You don't differentiate the sign function. The function $f(u) = \mbox{sgn}(u)\sqrt{2u}$ is simply an extension of $\sqrt{2u}$ to the entire real number line. For $u > 0$, differentiate this as $\sqrt{2u}$, and for $u < 0$, differentiate it as $-\sqrt{2u}$. The function is not differentiable at $u=0$.