I am currently reading Gilbarg-Trudinger's discussion of the eigenvalues of a self-adjoint scond-order elliptic operator.
Let $Lu = D_i(a^{ij}D_ju + b^iu) - b^iD_iu + cu$ with $a^{ij}$ symmetric. There is an associated quadratic form $\mathcal{L(u,u)}$ on the Hilbert space $H = W^{1,2}_0(\Omega)$: $$\mathcal{L(u,u)} = \int_\Omega(a^{ij}D_iuD_ju + 2b^iuD_iu + cu^2)dx$$ and the Rayleigh quotient is defined as $J[u] = \frac{\mathcal{L(u,u)}}{(u,u)_H}.$ Now define $\sigma = \inf_HJ.$ The text then claims that for all $v\in H$, $$\frac{d}{dt}J[u+tv]\bigg\rvert_{t=0} = 2(\mathcal{L(u,v)} - \sigma(u,v)_H)$$ How can this be proved?