Height to time equation: water cylinder with pinhole

In summary: JonathanIn summary, Jonathan found an equation for height as a function of time using the weight flow rate and cross sectional area of the container. He then integrated the equation to find the height at any given time.
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My inclination would be Physics Forums, Science Education, Homework and Coursework Questions, Calculus and Beyond

You have an analytic expression for the derivative of a function based on a fit to experimental data on the function itself. You know the uncertainty of the fit to the experimental data, and you would like to determine the uncertainty in the corresponding analytic expression for the derivative.
 

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