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
I would add to other comments that the specific derivative you mention (ax^b -> abx^(b-1)) is especially useful as if you look at any smooth curve and zoom in, it will start to look like the curve of some function ax^0 + bx^1 + cx^2 ….
Here’s an image to demonstrate [image](https://cocalc.com/blobs/projects/a8975d68-235e-4f21-8635-2051d699f504/.sage/temp/compute4-us/22699/tmp_zosLMe.svg?uuid=11530e5d-dd02-4f67-bdd3-4692f24b82bb)
The red line in the plot is a simple function of the form above which approximates the more complex blue one in the region of the marked point. The dotted line is an even simpler approximation of the same form.
No matter how complex the original function, we can use the fact it looks like this simpler function to find it’s derivative using the formula you gave. This allows us to do calculus with ‘real world’ functions that we can’t necessarily write a nice equation for and differentiate algebraicly.
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