At x=2, y is x^2 , which is 4. at x=3, y is 9. If y is changing at a rate of 2x for every change in x, wouldn’t that make y at x=3 6, because you moved 1 along the x so you move 2 along y? Or how does this work? I’m having trouble understanding differentiation 🙁
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A derivative is an *instantaneous* rate of change. The important concept to understand is that just because the derivative at x = 2 is 4, that doesn’t mean that this rate of change is “locked in” at 4 for all other values of x. The function y = x^2 doesn’t have a constant rate of change, but rather it increases faster and faster as x gets larger and more positive. Instead, when x is 2, the *instantaneous* rate of change, for that single point, is increase 4 y for every 1 x. But as soon as you go to a different x, even just a bit bigger than 2, the instantaneous rate of change also changes (and increases). And so on, for all values of x. By the time we get to x = 3, the instantaneous rate of change has increased from 4 to 6, which reflects the fact that x^2 increases faster and faster as x gets larger.
Here’s a real world example of the same idea: when you drive in your car and push down on the gas pedal, the car accelerates. At a particular moment during that acceleration, your *instantaneous* rate of acceleration might be an increase of 10 miles per hour, but a split moment later, your foot pushes down harder on the pedal and now your instantaneous rate of acceleration might be an increase of 20 miles per hour. It doesn’t stay the same because when you pushed down on the gas pedal, it caused a *smooth and continuous* acceleration, and at every point in time when you were speeding up, the instantaneous rate of acceleration was changing.
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