Finding approximation methods for a function and its derivative

Question

I need to find numerical methods to approximate a map $y=h(x)$ where $h: R^n \to R^m$ on the compact sets $D_x \times D_y$. More specifically, I need to find appropriate methods that ensure that, for every given $\varepsilon \geq 0$, $ | h(x) - \bar{h}(x) | \leq \varepsilon$ where $\bar{h}$ is the approximation of $h$. I am also interested in knowing the upper bound of the difference between their derivatives on the compact set, i.e. $| \frac{\partial h}{\partial x} - \frac{\partial \bar{h}}{\partial x} |$.