In our algebraic manipulations thus far of the Taylor series we have endeavored to obtain the first derivative and did what we had to do to eliminate the second derivative terms when we wanted a final result of second order. Here we simply notice that if we take the sum rather than the difference between and we can solve for the second derivative. We have
Finally, solving for the second derivative we find
This result is another of those which are widely used in computational physics simulations. Note that one needs three data values to obtain this second derivative. The centered character of the approximation has the benefit that it does not build in any directional bias. Of course, should one need a second derivative at one of the end points one would have to find a set of interior values which would also have the third derivative terms cancelling. We leave such a construction as an exercise for the reader.