I agree with you. Unless there's something else that we're missing here, the gradient of dotted with your is the positive radial derivative. Incidentally, the component of your gradient is incorrect; however, that's a moot point since you're taking the dot product of the gradient with the unit radial vector.
[EDIT] The OP has been corrected.
Question: in the book that has the minus sign, how do they define the unit normal vector? Is it some weird thing where it points inward or something? Is this sphere centered at the origin?
It's strange that they would intentionally make two mistakes in a row like that, unless there's some subtle aspect of the problem that we're missing here.
I don't either, directly, other than to say that it sort of looks like a chain rule. Your proof is much more direct. I would certainly agree that in cartesian coordinates, you haveI don’t get how to go from