If we numerically differentiate a given time series data consisting of *N* points by finite forward difference method, we will have N-1 points corresponding to first derivative. If it is a second derivative, we will have N-2 points and so on.


Let us say for the first derivative


$$
\approx\frac{f(x+\Delta x)-f(x)}{\Delta x}
$$


I have searched several books and webpages, but no one explicitly describes what is the corresponding x value for the numerically differentiated value if we wish to plot those *N-1* values.


In most science and engineering applications, we will not have an exact formula for f(x). One would use a set of data points ($x\_1$, $y\_1$), ($x\_2$, $y\_2$), . . . , ($x\_n$, $y\_n$) available to describe the functional dependence y = f(x). Many users ignore the $x\_1$ and use the remaining $x\_i$ for plotting N-1 differentiated points.


Others say that the first differentiated value corresponding to the average of $\frac{x\_1+x\_2}{2}$ i.e., this belongs to the center of $x\_1$ and $x\_2$.


What is the mathematically and rigorously correct way of dealing with $N-1$ values for the first derivative and $N-2$ values for the second derivative when we have N x-values? If we wish to plot them, how should we modify the x-coordinates?


**EDIT:** The reason for interest in the x-coordinates is utilitarian. The reason is that in chemical analysis applications, the derivative is used to locate the inflection points of titraton curve or detect a hidden peak in an over lapped spectrum. In such cases the interest is not in the accuracy of the value of the derivative but its corresponding x-coordinate. For example, in a potentiometric titration curve, the end point of titration is located by the first derivative, the inflection point's x-coordinate is the required volume.


Thanks.



Numerically differentiated values and their corresponding x-coordinates
The OP asks for a "reputable source", I would think that Press and Teukolsky's [Numerical Recipes](https://aip.scitation.org/doi/pdf/10.1063/1.4822971) [section 5.7 in [The Book](https://books.google.nl/books?id=1aAOdzK3FegC)] qualifies as such. As they explain, if you approximate $f'(x)\approx h^{-1}[f(x+h)-f(x)]$ the truncation error (from higher order terms in the Taylor expansion) is of first order in the small increment $h$. You can improve this to a truncation error of second order by symmetrizing, $f'(x)\approx (2h)^{-1}[f(x+h)-f(x-h)]$.


This can be readily generalized to higher order derivatives, just by repeatedly differentiating each term. The second derivative becomes
$$f''(x)=h^{-2}\left[f(x+h)+f(x-h)-2f(x)\right].$$


This is equivalent to $\frac{1}{4}h^{-2}[f(x+2h) + f(x-2h) - 2f(h)]$, as discussed at this [MSE question.](https://math.stackexchange.com/q/2064784/87355) 


