Error in a numerical derivative

Question

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.

Answer 1

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:

\begin{equation} 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). \end{equation}

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