# Error in a numerical derivative

I have a graph of data, say temperature ($T$) vs time($t$), I know the error bounds in each $\Delta T$.

The range of t is from 0 $\to$ 1600 s, with small steps say 0.001 s.

If I numerically take the derivative at some point on the graph, what is the error associated with that value.

The data is not suitable to be fit with a function or curve.

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For which times $t$ do you have corresponding values of $T$? And what do you mean when you say you know the error -- you mean you have some bound on the error? I doubt you know the error exactly. – littleO Nov 11 '12 at 11:06
yes I have error bounds, error bars on the graph. – John Echo Nov 11 '12 at 11:10
Ah. There are different methods for computing derivatives numerically. Do you know which method you want to use? – littleO Nov 11 '12 at 11:14
I'm using OriginLab8 and I think it takes the slope corresponding to 2-3 points. – John Echo Nov 11 '12 at 11:30

Here's a formula from Numerical Analysis by Burden and Faires (chapter 4.1).

\begin{align*} f'(x_0) &= \frac{f(x_0 + h) - f(x_0 - h)}{2h} - \frac{h^2}{6} f^{(3)}(\xi_0). \end{align*}

Notice that if the third derivative of $f$ is huge, the error might be huge.

There are other formulas for numerically computing derivatives, and they have similar expressions for the error. Here's one more example:

$$f'(x_0) = \frac{1}{12h}\left[ f(x_0 - 2h) - 8f(x_0 - h) + 8f(x_0 + h) - f(x_0 + 2h) \right] + \frac{h^4}{30} f^{(5)}(\xi).$$

If the fifth derivative of $f$ is huge, the error might be huge.

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