formal version of def. of derivative
i. f(c) is definiedii. lim x->c f(x) existsiii. lim x->c f(x) exists = f(c)
The formal definition of the derivative of a function f(x) at a point x=a is given by the limit:
f'(a) = lim (x -> a) [f(x) – f(a)] / (x – a)
provided that the limit exists. Here, f'(a) represents the derivative of f(x) at x=a, while the numerator in the above limit represents the difference between the values of the function f(x) at x and a, and the denominator represents the difference in the values of x and a. This limit represents the rate of change of the function f(x) with respect to its input x, at a specific point x=a.
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