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 🙁
In: 1
You’re not looking for the rate of change *between* 2 and 3, you’re looking for the rate *at* 2. You can find it by looking at smaller and smaller steps away from 2:
If you increase x from 2 to 3, y goes from 4 to 9, an increase of 5.
If you increase x from 2 to 2.5, y goes from 4 to 6.25, an increase of 2.25, or 4.5 for every x.
If you increase x from 2 to 2.1, y goes from 4 to 4.41, an increase of 0.41, or 4.1 for every x.
The smaller the increase, the more the ratio between the chance in y and the chance in x goes to 4. That limit is the derivative of x^2 at x=2. If you try the same at x=3, you will get a limit of 6. If you try it at any x, you will get a limit of 2x. That’s what is meant with “the derivative is 2x”.
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