I don’t mean in which cases I can use them, nor how they work. I know how they work (at least at a basic level, the derivative of ax\^b is abx\^(b-1), but I mean… why is a function that does those steps useful to solve any problem? It really seems like a random choice of operations.
In: Mathematics
The derivative of a function is its instantaneous rate of change. That’s often a very important property of a system.
If you have a moving object, the first two derivatives of its position are its velocity and acceleration. If you know some other property that relates to its velocity or acceleration, like the force on it, you’re on your way to something useful.
Consider a mass on a spring. As it moves back and forth, the force exerted by the spring depends on the compression of the spring, which depends on the position of the mass. So now you can relate the force (acceleration) to the position, which gives you a simple differential equation. You can solve the differential equation to describe how the mass moves, and you can even incorporate external forces and damping.
There are a whole bunch of situations where you know about how something changes, and you want to model that something over time. Temperature is another good one. If you have a heat source, you can figure out how fast heat is moving through the system, which is a rate of change. You want to know the temperature itself, so now you’re working with derivatives.
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