Ok so derivative are an expression of the rate of change of a function. Cool I get that.
– F(x) = 5 : the product of this function is always 5 there is no increase or decrease so there is no change no matter what X is and it makes sense that the derivative would equal 0.
– F(x) = 5x : it is obvious that each time x increases by 1 the product of this function increases by 5. I get it.
– F(x) = x² => F'(x) = 2x : starting from here the numbers stop matching and make me feel like I am missing something. F'(1) = 1. This makes perfect sense. F(x) did in fact increase by 1 when going from F(0) to F(1). Then I try F'(2) = 2×2 = 4. Huh ? But F(x) only increased by 3 between F(1) and F(2) ? Maybe I am looking at the rate of change as compared to F(0) ? after all there is an increase of 4 between F(0) and F(2). Let’s check with 3 then. F'(3) = 6. Wtf ?!
I don’t get it what does it mean when F'(2) = 4 ? When X = 2 then …? and what does it tell me about the original function. Thanks and hope my english isn’t too awfull.
In: 40
The thing you are missing is that derivatives measure the instantaneous rate of change, not an average rate of change. By average rate of change, I mean (F(1)-F(0))/1 is 1; the average rate of change from 0 to 1 is 1.
You are correct that if f(x)=x^2, then f(2)-f(1)=3 not 4. So the average rate of change from 1 to 2 is 3. But what you have missed is that f(3)-f(2)=5. The average rate of change on the other side of 2 is bigger! That is, the rate of change of this function increases as x increases.
Now, to make precise the notion of “instantaneous” rate of change goes a bit beyond ELI5, but I hope it makes sense that if x=1 to x=2 has an average rate of change of 3 and x=2 to x=3 has an average rate of change of 5, the instantaneous rate of change at x=2 should be in between those numbers!
EDIT: removed exclamation point; no factorials!
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