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.
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.