Comment on Can't forget unreasonable approximations either

• NoneOfUrBusiness@fedia.io ⁨19⁩ ⁨hours⁩ ago

  For the record you're presumably asking about derivatives (or, for the process, differentiation). A differential is just an infinitely small (infinitesimal is the term) change in some quantity. With that in mind, differentiation is how you take a function and get its rate of change (distance to speed is the usual example). You do this by essentially—and this is a theme with calculus—taking an approximation and making it so precise as to be exact. Specifically you take a small part of the curve and find its average rate of change, which is just the amount of change in the function divided by the amount of change in the independent variable. This is again an approximation, but if you make the part of the curve really really small the approximation becomes really really good until it's no longer an approximation. It's really just generalizing the idea of a slope to functions that aren't straight lines.

  Does that make sense?

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  • I_am_10_squirrels@beehaw.org ⁨14⁩ ⁨hours⁩ ago

    Exactly this.

    If you’re wondering why they’re useful, an example from my discipline is removing heat from a chemical reaction. You mix some ingredients together, stir for 2 hours, and you get a product. By combining the derivatives for the rate of reaction (how fast does A+B -> C) and the heat of reaction (how much energy does the reaction release), you can calculate how much cooling you need to prevent a runaway reaction that takes out half the neighborhood.

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