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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What you’re describing in your third bullet-point is a line SECANT to the parabola that intersects it at exactly two points, at F(1) = 1 and F(2) = 4, and it has a slope = 3.
The derivative is the value of the line TANGENT to parabola at that point, that intersects is exactly once, THAT is what has has a slope = 4 in your example.
In your example, the tangent line is steeper than the secant line, that’s why it has the higher value. You can’t add and subtract values of a function and its derivative the way you were, that’s why the numbers aren’t lining up!
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