Return to Answer

The OP asks for a "reputable source", I would think that Press and Teukolsky's Numerical Recipes [section 5.7 in The Book] 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 MSE question.

The OP asks for a "reputable source", I would think that Press and Teukolsky's Numerical Recipes [section 5.7 in The Book] 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.}

The OP asks for a "reputable source", I would think that Press and Teukolsky's Numerical Recipes [section 5.7 in The Book] 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, $f'(x)\approx (2h)^{-1}[f(x+h)-f(x-h)]$.

The OP asks for a "reputable source", I would think that Press and Teukolsky's Numerical Recipes [section 5.7 in The Book] 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 MSE question.

The OP asks for a "reputable source", I would think that Press and Teukolsky's Numerical Recipes 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, $f'(x)\approx (2h)^{-1}[f(x+h)-f(x-h)]$.

The OP asks for a "reputable source", I would think that Press and Teukolsky's Numerical Recipes [section 5.7 in The Book] 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, $f'(x)\approx (2h)^{-1}[f(x+h)-f(x-h)]$.