How do you calculate the gradient of a scalar product? I am trying to follow the calculation that yields
$$\nabla\langle x, Ax \rangle=2Ax$$
For a symmetric, real matrix $A$.
Do I use the bilinearity of the scalar product? The product rule for the gradient? I do not know where to start and what is legitimate as I am unexperienced with these symbols.

Simply without any coordinates just with Leibniz for any inner product:
$$d_p(\langle x,Ax\rangle)=\langle p,Ax\rangle+\langle x,Ap\rangle
=\langle Ax,p\rangle+\langle A^\star x,p\rangle=\langle(A+A^\star)x,p\rangle,$$
where $A^\star$ denotes the adjoint (see https://en.wikipedia.org/wiki/Hermitian_adjoint) of $A$.
