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I've been searching the internet for a few weeks and picking the brains of colleagues in person without success. As to what happens to error propagating down a derivative when the error in a measurement is known. (Hopefully I'm on the right SE, Physics was also tempting for the question)

In my work, I have a monotonically decreasing function $f(x,y)$ where $x(t), y(t)$ are functions of time which also decrease with time. I know the error in each $x$ and $y$ to be $\delta x, \delta y$, and then the error in $f$ is $\delta f = \sqrt{(\frac{df}{dx})^2 \delta x^2 + (\frac{df}{dy})^2 \delta y^2}$.

However, $\frac{df}{dt}$, has to have some error associated with it, that corresponds intuitively with the error in the original measurements $\delta x, \delta y \rightarrow \delta(\frac{df}{dt})$. The error in the derivative wrt of $f$ is desired, $\delta(\frac{df}{dt})=$?

First attempts to figure out what $\delta(\frac{df}{dt})$ are have not been successful. I always end up stuck with an unknown error of some other time derivative by standard error propagation rules. I've also tried taking the derivative wrt of $\delta f$, but this value is not monotonic, so it's derivative is not always positive and I would reason that my error (essentially a standard deviation) should be a positive number.

enter image description here

Here's some example data that represents the issue at hand of $x(t), y(t), f(x(t),y(t))$. Here $f=\frac{\pi}{6} (3 y^2 x + x^3)$ and the bands represent the error in the measurements. The error in $y$ is constant, while the error in $x$ varies with time, and the influences of these on $f$ give an overall narrowing error band.

Brandon Murray
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    You might get more insight, rather than thinking about "error" as a number, think of your measurements as intervals. So the measurement pair $x(t), y(t)$ gives the rectangle $(x(t) - \delta x(t), x(t) + \delta x(t)) \times (y(t) - \delta y(t), y(t) + \delta y(t))$. Then you evaluate $f$ at each point of that rectangle, record the min, the max, and their midpoint. From those, get the $f$ and $\delta f$. But the distribution of $f$s is not usually uniform or symmetric; more honest is to think of $x(t)$ and $y(t)$ as distributions, and obtain a distribution for $f$. – Eric Towers Oct 01 '19 at 02:10
  • Thanks - So I think I may have over complicated things by bringing in $x$ and $y$. When you say distributions for $x$ and $y$, you mean each unique point in $t$ could be a range in values an I just happened to have 1 value off of a Gaussian distribution recorded? If so, then I see how this could get me towards $f$, but I'm a bit lost how to extend this to the derivative of $f$. I'll add a plot to the original post with $x, y,$ and $f$ – Brandon Murray Oct 01 '19 at 12:28

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