Derivative of the Exponential map explained

Question

I'm trying to understand how to interpret the derivative of the exponential map in terms of the Jacobi field? Define the map $$ f(x):Exp_p(x), $$ where the Riemannian manifold is on $\mathbb{R}^d$ with a non-Euclidean Riemannian metric making it geodesically complete.
Is there a nice expression for it's gradient?