Several mean value theorems for higher order divided differences and approximate Peano derivatives are proved.

For a differentiable function $𝐟:I\to {ℝ}^k,$ where $I$ is a real interval and $k\in \mathbb{N}$, a counterpart of the Lagrange mean-value theorem is presented. Necessary and sufficient conditions for the existence of a mean $M:I^2\to I$ such that$$𝐟\left(x\right)-𝐟\left(y\right)=(x-y){𝐟}^{\text{'}}\left(M(x,y)\right),x,y\in I,$$ are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented.

For subspaces, $X$ and $Y$, of the space, $D$, of all derivatives $M(X,Y)$ denotes the set of all $g\in D$ such that $fg\in Y$ for all $f\in X$. Subspaces of $D$ are defined depending on a parameter $p\in [0,\infty ]$. In Section 6, $M(X,D)$ is determined for each of these subspaces and in Section 7, $M(X,Y)$ is found for $X$ and $Y$ any of these subspaces. In Section 3, $M(X,D)$ is determined for other spaces of functions on $[0,1]$ related to continuity and higher order differentiation.