Suppose $x$ is a finite discrete signal consisting of values of a continuous signal $X(t)$ measured at equal intervals, say with sampling frequency $f_s$. So the time step size is given by $h=f_s^{-1}$. In other words, $x_n=X(t_n)$ with $t_n=t_0+n\cdot h$ for $n=0,1,2,\ldots N$ and some natural number $N$.

The central difference formula states that the derivative $x'_n=X'(t_n)$ for $0<n<N$ is approximated by: $$x'_n=\frac{x_{n+1}-x_{n-1}}{2h}=\frac{f_s}{2}(x_{n+1}-x_{n-1})$$ Let $w$ be the filter defined by $$w_n=\left\{\begin{array}{ll} n & \mathrm{if\;}|n|=1\\ 0 & \mathrm{otherwise}\end{array}\right.$$ In a finite vectorial description we choose $w=(-1\;\;0\;\;1)$. When we extend the measurement data with zeros on the left- and right-hand side. if necessary, we can als write the formula for the approximation of the first derivative in the measurement points in terms of a cross correlation: $$x' = \frac{f_s}{2}\cdot x \otimes w$$ When a programming language has built-in functions for these operations, we can use them to calculate first derivatives. If you don't want to extend measurement data with zeros, then you have to be happy with a derivative in fewer points than you measured because the edge points in the dataset cannot be handled in this case.