Both $f'$ and $f^\prime$ can be used for derivatives in LaTeX. However, it seems that these expressions are not treated as equivalent in Approach0.

Here is an example: the queries $|f(z)+zf^\prime(z)|<1$ and $|f(z)+zf'(z)|<1$ give different results.

Has this been observed/reported somewhere already? Otherwise I would offer to add it as another tip to How to search on this site: Approach0.

Looking at the source posted on GitHub I see the following:

"'"                                { RET_TOK(prime, PRIME, WC_NONCOM_OPERATOR, PRIME); }
\\prime                              { RET_TOK(_AUTOGEN_, PRIME, WC_NORMAL_LEAF, VAR); }

Without knowing exactly what RET_TOK does, at least I can se that both of them have _token=PRIME.

Of course, you're asking more about ^\prime than just \prime.