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How can we use Taylor series to derive the truncation error of the approximation

$f^\prime(x)\approx\frac{f(x+h)-f(x-h)}{2h}$

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Hint: Start by taking the Taylor polynomial of degree 2 for $f(x+h)$ (as a function of $h$), and then substitute $-h$ for $h$ in this...

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    @Sunday: You lose some accuracy, if you don't take advantage of the $h\leftrightarrow-h$ symmetry of the R.H.S. As a toy example compute both $(f(x+h)-f(x-h))/2h$ and $(f(x+h)-f(x))/h$ at the local minimum of $f(x)=x^2$.2011-07-05