# Uncertainty analysis of discrete derivative

Let's take evaporation case as a example. A water tank is mounted on a scale (resolution-$1g$) for evaporation. The scale auto record the result every $1s$. If the value changes at time $T$, evaporation can be calculated by:

$$e_{t}=\frac{(t_{2}-t_{1})}{T}\tag{1}$$

To do the uncertainty analysis, we can get full derivative of $e_t$

$$\mathrm{d}(e_t)= \frac{\mathrm{d}e_t}{\mathrm{d}t_{1}}(1g)+ \frac{\mathrm{d}e_t}{\mathrm{d}t_{2}}(1g) + \frac{\mathrm{d}e_t}{\mathrm{d}T}(1s).$$

However, using FDE method (equation 1) makes a lot of noise to the $e_t$ data. to smooth out, we can use spline interpolation on $e_t$ readings (ie. use spline interpolation on the scale data over time, then use equation 1 to work out the $e_t$, this method would reduce the noise significantly). However, after interpolation, can we still use this kind of method to work out the uncertainty of the $e_t$?

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