I was just wondering in anyone's experience, if they came across an official WolframAlpha Calculation (using https://www.wolframalpha.com/) that was different from one of the widgets (which I think is unofficial and community made, for example, this).
The issue is, I don't understand if they are actually giving me different answers (thus causing me to cast doubt on ever using WolframAlpha Widgets again unless I am completely wrong on this).
- WolframAlpha Official
If they aren't could someone tell me if these expressions are mathematically equivalent?
The two answers are identical.
Wolfram Alpha uses Wolfram Language as the underlying syntax. The correct expression in Wolfram Language would be
leading to the same answer as in the widget.
My guess of why you are getting the same answer expressed differently is because there is ambiguity in your Wolfram Alpha Official input. In the input
WAO doesn't immediately know if what you want is
or
or one of the two above inputs, but with respect to y or to rho. Therefore it has to guess what is the most likely thing that you want - presumably this involves some internal rearrangements of the input, and once WAO figures out what it is that you most likely want, it performs the derivative with respect to the input that WAO thinks is correct.
If you clarify your intent by enclosing the expression in brackets and specifying the variable with respect to which the derivative should be computed, e.g.
the output will be identical to that of the widget.
Note that the widget doesn't have these ambiguities because from f(x,y)=expression it knows what the variables are (because they're specified in f(x,y)), and it knows what the intended expression is.