Alternate definition of derivative
limit as x approaches a of [f(x)-f(a)]/(x-a)
One alternate definition of derivative is the instantaneous rate of change of a function at a specific point. It can also be interpreted as the slope of the tangent line to the function at that point. This definition highlights the idea that the derivative captures the behavior of a function very close to a specific point, rather than over a larger interval. It is often represented using the limit definition:
f'(x) = lim(h -> 0) [f(x + h) – f(x)] / h
where f'(x) is the derivative of f at x, h is a small change in x, and the limit as h approaches zero represents the instantaneous rate of change of f at x.
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