How to estimate the norm of Jacobi fields or to compute $\langle J(0),J'(0)\rangle$

Question

There is a classical formula based on Taylor series for the square norm of a Jacobi field $J$ with $J(0) = 0$ along a geodesic on a Riemannian manifold $M$. I am interested on a possible estimate for the square norm of a Jacobi field $J$ such that $J(0) = X\neq 0$. Or simply, it would be nice to understand the relation between $J(0)$ and $J'(0)$. Can we guarantee under any assumption that $\langle J(0),J'(0)\rangle \neq 0$?