limit as x approaches a of [f(x)-f(a)]/(x-a)
Alternate definition of derivative
This is the formula for the definition of the derivative of a function f(x) at a point a. Thus, the answer to this question is the derivative of f(x) evaluated at x=a.
The derivative measures the instantaneous rate of change of a function at a specific point and is defined as the limit of the average rate of change as x approaches a. In other words, as x gets arbitrarily close to a, the average rate of change between f(x) and f(a) divided by (x-a) approaches a single value which is equal to the slope of the tangent line to the curve of f(x) at the point x=a.
Therefore, the answer to this question is the derivative of f(x) at x=a. Written in mathematical notation, it is:
f'(a) = lim(x -> a) [f(x) – f(a)]/(x-a)
Note that this formula only applies if the limit exists and is finite. If the limit does not exist or is infinite, then the derivative does not exist at x=a.
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