I have a simple question regarding the differentiation of the logarithm mapping in Riemannian manifolds:

Assume that $M$ is a compact Riemannian manifold, isometrically embedded into $\mathbb{R}^n$.

Let $p(t), q(t): [0,1] \to M$ be two curves.

I am interested in bounding the quantity

$$ \frac{d}{dt}\log_{p(t)}(q(t)), $$

more precisely, does the estimate $$ \left|\frac{d}{dt}\log_{p(t)}(q(t))\right|\lesssim \left|\frac{d}{dt}p(t) - \frac{d}{dt}q(t)\right| + \left|p(t) - q(t)\right| $$ hold true, at least if $p(t)$ and $q(t)$ are close to each other??