Let $(M^n,g(t))$, $t\in[0,T]$, be a smooth Ricci flow. Assume that $\overline{B}_{g(0)}(x_0,r)$ is compactly contained in $M$ and that \begin{align*} |\operatorname{Rm}|\le K \end{align*} on $B_{g(0)}(x_0,r)\times[0,T]$. For each integer $m\ge 1$ and each $0<\tau\le T$, there is a constant $C=C(n,m,K,r,T,\tau)>0$ such that \begin{align*} |\nabla^m\operatorname{Rm}|\le C \end{align*} on $B_{g(0)}(x_0,r/2)\times[\tau,T]$. If the flow is complete and $|\operatorname{Rm}|\le K$ on all of $M\times[0,T]$, then the positive-time estimate has the scale-invariant form \begin{align*} |\nabla^m\operatorname{Rm}|\le C_m K\left(t^{-m/2}+K^{m/2}\right) \end{align*} for $0<t\le T$, where $C_m$ depends only on $n$ and $m$. In particular, for $0<t\le \min\{T,K^{-1}\}$ this gives $|\nabla^m\operatorname{Rm}|\le 2C_mK t^{-m/2}$ after increasing the constant.