Calculating with Method of Equal-area differentiation

I have been trying to understand this method although I admit, don’t have a very good calculus background just yet.
For a given reaction where $x$ is time, $f(x)$ is cell concentration and assuming it to be first order (autocatalytic), we obtain the table:
\begin{array}{rrrrr}\hline
x & 
f(x) &
\frac{\Delta f}{\Delta x} &
\frac{\mathrm{d}f}{\mathrm{d}x} &
\text{actual}\frac{\mathrm{d}f}{\mathrm{d}x}\\\hline
0 & 0 & - & 1010 & 1000 \\
0.2 & 182 & 910 & 805 & 818 \\
0.4 & 332 & 740 & 670 & 670 \\
0.6 & 451 & 605 & 550 & 548 \\\hline
\end{array}



For this data: Is it possible to calculate the $\mathrm{d}f/\mathrm{d}x$ column with the rate formula? Or maybe $\mathrm{d}f/\mathrm{d}x$ can only be calculated by using the equal areas given in the plot? 


