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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The derivative is not 2dx, it is 2x. Your example is using the former. If we go from 2 to 3 then we need to add 2x to y and x starts at 2. So 2^2 + 2×2 = 8 which is pretty close to 9.
Now by the time we reach x=3 the derivative has gone up to six so to be more accurate we can use the value of the derivative that is between x=2 and 3 for the whole trip. The average of 4 and 6 is 5. Using 5 as the derivative we can go from x=2 to x=3 as 2^2 + 5 = 9. This time we arrived at exactly 3^2!
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