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 example you gave is specifically for polynomials. Those steps are only useful because, with specifically polynomials, this represents the rate of change in that polynomial.
There is no ax^b for a function like sin(x) yet its derivative is way *more* useful.
There is a broader test to use for derivatives, and this involves taking the change over time (rise over run) as the run gets smaller. What does the rise over run get closer and closer to as the run gets closer to zero? This definition tells us, more or less, the slope of the line at a point. How one number changes in response to another number. That’s important anywhere that change is happening.
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