YES!

Or, more correctly,

**rate** of change, not amount of variation.

NO!

That is not how the derivative is defined. "2 distinct points" give a

*average* rate of change between those two points. The derivative requires the

*limit* that as the distance between the two points. Didn't your Calculus I textbook, if not you teacher, emphasize the limit concept? That is the crucial idea that defines "Calculus" and makes it different from "Algebra"!

That's odd. I don't know who you mean by "they" but every textbook I have seen has made it clear that the "derivative" is the "instantaneous rate of change" or "slope of the tangent line". In fact, you just used the phrase "slope of the tangent line" yourself, above. Aren't you clear on what that means? Do you know what the "slope of a line" tells you? Do you know what the tangent line to a curve, at a point, is?

If the definitions in Calculus texts aren't good enough, I am not sure what kind of definition you want. The derivative of a function tells you the instantaneous rate of change. If you know that some object has moved from x= 0, at time t= 0, to x= 10, at time t= 1, then you know that the

**average** speed (rate of change of position) is (10- 0)/(1- 0)= 10. If you know that the "position function" is x= 10t^2, then you know that, at each time t, between 0 and 1, the instantaneous speed, the speed at

**that instant**, not over some span of time, was dx/dt= 20t.

And, I'm Brazilian, so, sorry in advance for possible typos, thank you in advance