After 28 years of (amateurishly and rather clumsily) programming computers and 25 years of using calculus, I just found out about a method to calculate numerical derivatives exactly (up to numerical accuracy.)
I knew how to do numerical differentiation based on approximating derivatives with finite differences which incurs inexactness due to truncating infinite series, but I never thought about how to get exact numerical results. Luckily someone else did, and we got the automatic differentiation technique.
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