covariant derivative of manifold-valued function and logarithm map

Question

Let $M$ be a Riemannian manifold and $f\colon \Omega\subset \mathbb{R}^d\rightarrow M$ a smooth, i.e. $C^\infty$, function. For any $p\in M$ let $T_pM$ be the tangent space at $p$ and $\log_p\colon U\subset M\rightarrow T_pM$ the logarithm map, i.e. the local inverse of the exponential map. Note that the function $\log_p\circ f$, wherever its defined, is a map between linear vectorspaces. Hence we can consider the classical derivatives (first and higher order) of this composition. I would like to estimate these derivatives in terms of derivatives of $f$ and $\log_p$, e.g. covariant derivatives. In a first case it would be sufficient to estimate derivatives at $x$ for the special case $p=f(x)$, i.e. $D^v \log_p \circ f(x)|_{p=f(x)}$. Then I would like to have an inequality of the form $|D^v \log_p \circ f(x)|\leq |D^v \log_q \circ f(x)|_{q=f(x)}|+C(f)d(p,f(x))$ where $C$ is a constant depending only on $f$, e.g. the norms of the covariant derivative of $f$. I would like to have such estimates to bound the error of an approximation operator for manifold-valued function.