Taylor expansions
With these Taylor expansions the approximations for the derivatives takes the form
$$
\begin{align}
&\left[\frac{\partial u(x,t)}{\partial t}\right]_{\text{approx}} =\frac{\partial u(x,t)}{\partial t}+\mathcal{O}(\Delta t) ,
\tag{12}\\ \nonumber
&\left[\frac{\partial^2 u(x,t)}{\partial x^2}\right]_{\text{approx}}=\frac{\partial^2 u(x,t)}{\partial x^2}+\mathcal{O}(\Delta x^2).
\end{align}
$$
It is easy to convince oneself that the backward Euler method must have the same truncation errors as the forward Euler scheme.