The OP asks for a "reputable source", I would think that Press and Teukolsky's <A HREF="https://aip.scitation.org/doi/pdf/10.1063/1.4822971">Numerical Recipes</A> [section 5.7 in <A HREF="https://books.google.nl/books?id=1aAOdzK3FegC">The Book</A>] 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 discussed in some detail at this <A HREF="https://math.stackexchange.com/q/2064784/87355">MSE question.</A>