Why didn't we define the derivative matrix on a grid through the familiar asymmetric derivative: [begin{equation*}frac{d f(x)}{d x}=lim _{Delta x ightarrow 0} frac{f(x+Delta x)-f(x)}{Delta

Why didn't we define the derivative matrix on a grid through the familiar "asymmetric" derivative:

\[\begin{equation*}\frac{d f(x)}{d x}=\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} ? \tag{2.90}\end{equation*}\]

What are its eigenvalues? What happens when it is exponentiated?