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## Main Question or Discussion Point

Google has failed me. Any responses are greatly appreciated.

- Thread starter epicfailguy
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Google has failed me. Any responses are greatly appreciated.

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- #3

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f(x+t) = exp(t d/dx) f(x)

The symmetric difference with step t is thus given by:

Delta_t f(x) = [exp(t d/dx) - exp(-td/dx)]/2 f(x) = sinh(t d/dx) f(x)

So, this means that formally we can express the derivative operator in terms of the finite symmetric difference operator as:

d/dx = 1/t arcsinh(Delta_t) = 1/t [Delta_t - 1/6 Delta_t^3 + ...]

So, to comnpute the derivative at a point, all you need to do is to repeatedly apply the symmteric finite difference operator with some stepsize t. The smaler you take t, the faster te series converges, but then you lose significant digits. So, you should take t not too small and a few terms of the series.

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jhae2.718

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Ref: Calculus: Graphical, Numerical, Algerbraic, by Ross Finney et. al. p. 111.

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