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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I mean I think the other answers covered it but the thing that I always visualized was the tangent line. If you have a linear equation like F(x) = 5x.. then it doesn’t really make sense to draw a tangent line of course because it will be the same line.. but if you did that, the slope of the line would be 5. So the slope of the tangent line is your derivative (5).
Now, for a parabola, the tangent line is different at any given point, and gets steeper as the parabola trends upward. But the slope of that line will be your derivative. For the x^2 example, of course the derivative will always be 2x.. or put another way, the slope of the tangent line will always be 2x.
The reason I bring up the tangent line is because the tangent line only touches the parabola at a single point and is a different line with a different slope at every point. This drives the ‘point’ home that the derivative is about singular points along the line, not big jumps like from F(1) to F(2).
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