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.