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What does it mean by the rate at which y changes as x changes?
The derivative of x^2+x is 2x+1, which is the rate at which y changes as x changes. But if you plug in 1 and 2 for x in the original equation, you get a change of 6 which is not the change of 2 in the derivative equation?
1 Answer
- PopeLv 76 years agoFavorite Answer
This is an important leap for you to make in calculus. The derivative is the instantaneous rate of change of the function. You (I believe) have tried to compare it to the average rate of change over an interval.
Let g(x) = x² + x.
g'(x) = 2x + 1
g(1) = 2
g(2) = 6
From x = 1 to x = 2, g(x) has increased by 4 (not 6). Its average rate of change over that interval is 4/1 = 4.
g'(1) = 3
g'(2) = 5
This means that at x = 1, g(x) has a certain rate of change, and at x = 2, it has a greater rate of change. The average rate of change from 1 to 2 does not have to equal either of these values, and in this case it does not.
