Convert to camelCase
Retina, 56 bytes

Byte count assumes ISO 8859-1 encoding.

T`'\`
S_`\W|_|(?&lt;=[a-z])(?=[A-Z][a-z])
T`L`l
T`l`L`¶.
¶



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Explanation

This implements the specification quite literally:

T`'\`


Remove apostrophes and backticks.

S_`\W|_|(?&lt;=[a-z])(?=[A-Z][a-z])


Split the string around non-word characters (in regex this also excludes digits and underscores), or underscores or positions that have a lower case letter on the left and and upper case, lower case on the right. This would create some empty segments when there are two non-letter, non-digit characters in a row, or more important at the beginning of the string. We get rid of those with the _ option. Here, "splitting" means put each remaining part on its own line.

T`L`l


Convert everything to lower case.

T`l`L`¶.


Convert each character that occurs after the linefeed to upper case. This will conveniently skip the first word because there's no linefeed in front of it.

¶



Get rid of the linefeeds to join everything back together.