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Your solution seems ok to me, though you might want to complete it by proving it for spherical coordinates as well. Its practically the same as this, since in the Del operator in spherical coordinates you will see that all the non radial derivatives disappear as well.
after all, radial usually means spherically radial, since cylindrically ...
A function is shown to be irrotational. The OTA has produced a different method of proof than the one that was included with the question. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.