I have a series $x$ and a function $f(x)$ and I need to calculate the scalar differences $\operatorname{diff}(f(x))$.
Having both $x$ and the function $f$ available, I can simply replace $x$ in $f$ and obtain the data I need. But wanting to use the analytic function of the derivative $d(x)(f)$ which relationship exists between $\operatorname{diff}(f(x))$ and $d(x)(f)$?
Or should I use a different derivative function?
The important thing that interests me is that I don't have to move on the $x$-axis for the calculation
as is the case with $\operatorname{diff}(f(x_i) - f(x_i-1))$.
So for each $x_i$ I obtain $d(x)(f(x_i))$ using only $x_i$.
Basically I'm asking for the inverse relation of the finite difference method. In my case I want to calculate the approximation of finite differences given the derivative function. The step of the differences is not known but I have the differences of $x$ available to compute the differences of $f(x)$ using the derivative of $f$.
Thanks!
