/sci/ - Science & Math

>>16230358
anon surely you learned what a slope is in school? here are some ways to think about linearity
straight line: it's in the name
constant slope: the slope of a linear function is the same everywhere on its domain (in other words, the derivative of the function is constant) This is the same thing you were arguing with the change thing for (x+Δx)^2, just in generally useful language.
it's a linear transformation: the function f satisfies f(x+y) = f(x) + f(y) and f(xy) = xf(y) (or of the form g(x) = f(x) + C for f a linear transformation and C a constant, in order to allow lines not through the origin) cf >>16230380. This might be a useful exercise to you: prove that all linear transformations from the real line to itself is of the form f(x) = mx for some constant m (hint: what is f(0), what is f(1)?)
the exponent of "variables" are 1: cf >>16230379
for all of these notions the only linear functions are the ones of the form mx+c

>>16230492
>the other end takes for granted that every derivate has its own logical explanation
the derivative does have its own logical explanation: the derivative of f at x is the slope of f at x. A function is linear if and only if its derivative is a constant function.
>what does 1/x actually mean to have a decreasing derivative?
what do you mean decreasing derivative? it's not constant anyways and that's what matters