Difference in differentiation between Mathematica and Wolfram Alpha

Question

I am trying to differentiate this: $$f(x)=e^{-x^2}$$

In Wolfram Alpha I get this: $$-2x\,e^{-x^2}$$

But in Mathematica I get: $$-2x\,e^{-x^2}\log(e)$$

Why the difference?

Answer 1

The difference is in notation.

Mathematica, by convention, only assigns special meaning to identifiers that begin with a capital letter, not lowercase. In particular, the mathematical constant $e$ is represented as E in Mathematica - when you type it in. When it appears in output, it is written as "" to distinguish it.

A lowercase e in Mathematica has no special meaning: it is a variable, just like f or g. Mathematica treats it as a variable: you can tell because of the syntax highlighting. If e were equal to E, then Log[e] would simplify to 1 and go away.