In the code that my group is writing (Lethe) we use a stabilized approach to solve the Navier-Stokes equation. The GLS stabilized method we use has a stabilisation term which contains a stabilization parameter such as: $$\tau =\left[\left( \frac{2\rho |\mathbf{u}|}{h} \right)^2+\left( \frac{4 \mu}{\gamma h^2} \right) \right]^{0.5}$$
where $\gamma$ is a constant that depends on the type of element, $\rho$ is the density, $h$ is the element size and $\mu$ is the dynamic viscosity.
This stabilization parameter does not depend on the flow variable except for the first term which contains the absolute value of the velocity. I find myself unable to calculate the jacobian of that term, so right now we assemble an analytical jacobian for everything but this term.
Is there anyway to proceed to establish the jacobian of this term analytically? I understand that the $u^2$ term can be treated straightforwardly, but I am unsure on how to calculate the Frechet derivative of the square root.