Alternative form of the definition of the derivative
lim x->c. f(x)-f(c) / x-c
The alternative form of the definition of the derivative is:
f'(a) = lim(x->a) [f(x) – f(a)] / [x – a]
This form of the definition is also known as the symmetric difference quotient, as it involves taking the difference from both sides of the point a. The numerator is the change in the function value between x and a, and the denominator is the change in x. Then, we take the limit as x approaches a to find the instantaneous rate of change at the point a.
This definition is equivalent to the traditional definition of the derivative, f'(a) = lim(h->0) [f(a + h) – f(a)] / h, where we let h represent the small change in x from the point a, and take the limit as h approaches 0. The symmetric difference quotient is useful in certain situations where we are dealing with unevenly spaced data or functions that are not smooth and differentiable.
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