Consider a constant speed curve along a meridian of the revolved surface S,
where is some open interval of .
To show is a geodesic, you want to show that is orthogonal to S.
Since it has constant speed, you can say something about the relationship between and .
Consider the plane P that defines the meridian by cutting through S.
Since stays in P, so do all of its derivatives.
At any point along , P is spanned by and the normal to S at that point.
Put it all together, and you deduce something about that proves that is a geodesic.