[scythe]

The function 1^x is constant everywhere. You can write (-1)^x to get the kind of effect you are looking for, but doing this with 1^x is madness.

>The only disadvantage of this approach is that symbolic differentiation and integration are more complicated, by multiplication with a constant.

There is more to it than that. The exponential function is not really e^x, it is lim_{N->∞} (1 + x/N)^N. The advantage of this definition is that, since it is valid everywhere on the complex plane, it allows the rigorous definition of real powers without recourse to inverse functions. We still teach students to prove things.

But also, isn't this a little contradictory? The goal of the original blog post was to simplify various differential equations. If you aren't simplifying symbolic differentiation, then what are you simplifying?