Solution of Mumford's second problem on theta functions

A complete solution of Mumford's second problem about representation of theta derivatives with rational characteristics in terms of theta constants with rational characteristics is found. An explicit formula for computing such an expression for theta derivative with an arbitrary rational characteristic is derived, and illustrated with examples. This approach fails only on four characteristics: [(1/2)(1/6)], [(1/6)(1/2)], [(1/6)(1/6)] and [(1/6)(5/6)]; however expressions for theta derivatives with these four characteristics are found by other authors. Expressions for theta derivatives appear to be homogeneous of degree 3 with respect to theta constants.