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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Think about it this way. Lets say you are in a rocket, and to measure your speed you start by measuring distance in meters from where you started. Coincidentally let’s say it’s a 10 second test, where your distance is defined by t². So at 1 second your 1 meter away, 2 your 4 meters, 3 your 9 meters away, etc. At the end of the 10 seconds your 100m away.
That’s great but your trying to measure your speed each second. So first you say well I went 100 meters in 10 seconds so 10 meters per second. ….But wait that’s not your top speed, your top speed is 20m/s.
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