$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Newton's method (1)
Need to evaluate \( F_i(u^{-}) \):
$$
F_i \approx \hat F_i =
\int_\Omega (u^{-}\baspsi_i + \Delta t\,\dfc(u^{-})
\nabla u^{-}\cdot\nabla \baspsi_i
- \Delta t f(u^{-})\baspsi_i - u^{(1)}\baspsi_i)\dx
$$
To compute the Jacobian we need
$$
\begin{align*}
\frac{\partial u}{\partial c_j} &= \sum_k\frac{\partial}{\partial c_j}
c_k\baspsi_k = \baspsi_j\\
\frac{\partial \nabla u}{\partial c_j} &= \sum_k\frac{\partial}{\partial c_j}
c_k\nabla \baspsi_k = \nabla \baspsi_j
\end{align*}
$$
