$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Leading-order error terms in finite differences (1)
$$
\begin{align}
\lbrack D_tu \rbrack^n &= \frac{u^{n+\half} - u^{n-\half}}{\Delta t} = u'(t_n) + R^n
\tag{3},\\
R^n &= \frac{1}{24}u'''(t_n)\Delta t^2 + \Oof{\Delta t^4}
\tag{4}\\
\lbrack D_{2t}u \rbrack^n &= \frac{u^{n+1} - u^{n-1}}{2\Delta t} = u'(t_n) + R^n
\tag{5},\\
R^n &= \frac{1}{6}u'''(t_n)\Delta t^2 + \Oof{\Delta t^4}
\tag{6}\\
\lbrack D_t^-u \rbrack^n &= \frac{u^{n} - u^{n-1}}{\Delta t} = u'(t_n) + R^n
\tag{7},\\
R^n &= -{\half}u''(t_n)\Delta t + \Oof{\Delta t^2}
\tag{8}\\
\lbrack D_t^+u \rbrack^n &= \frac{u^{n+1} - u^{n}}{\Delta t} = u'(t_n) + R^n
\tag{9},\\
R^n &= {\half}u''(t_n)\Delta t + \Oof{\Delta t^2}
\tag{10}
\end{align}
$$
