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
How much did F(x) increase between 1 and 2? 3.
How much between 1.9 and 2? 0.39. But we should adjust for the smaller change in x, by dividing by 2 – 1.9 (= 0.1), to get 3.9.
How much between 1.99 and 2? (Make the right adjustment.) 3.99
How much for between 1.999 and 2? 3.999.
Notice a pattern here? What happens as we take smaller and smaller intervals to analyze? The answer narrows down on a certain value. That’s what the derivative of a function is.
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