How to compute this derivative?

Question

Let $H(y)$ be a linear operator and $H^*$ it's adjoint. \begin{align} \mathcal{L} = 1/2* \Vert x + H^*(u u^\mathsf{T})\Vert^2 \end{align} how to compute \begin{align} \frac{\partial \mathcal{L}}{\partial u} = 1/2* \frac{\partial }{\partial u} \{\Vert x + H^*(u u^\mathsf{T})\Vert^2\} \end{align}