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.
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f(x) = x^2 is the first function in your examples that becomes curved – it’s no longer a straight line. So what changes from that? What changes is that when you go from x = 1 to x = 2 the rate of change “ramps up”. When you’re near x = 1 it’s still approx. 1, but at x = 1.1 it’s 1.21. In this way it ramps up all the way to 4 at x = 2.
So what can you learn from this? You should notice that there’s also lots of numbers, in fact infinite amount, between the numbers you pick. And they all have their own rates of change for such non-linear or curved functions as f(x) = x^2. I’d like you to correct your intuition about derivatives from them being a straight line between any two arbitrary picked numbers like you did, to instead it being a straight tangential line that merely touches the function at just a SINGLE point as far as you’re concerned.
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