Alternative form of the definition of the derivative
The derivative of a function f(x) at a specific point x=a can be defined in an alternative form using the concept of limits
The derivative of a function f(x) at a specific point x=a can be defined in an alternative form using the concept of limits.
The alternative form is expressed as:
f'(a) = lim h->0 [f(a+h) – f(a)] / h
In this form, we consider the difference between the values of the function at the point a and the point a+h, where h represents a small increment in the value of x. This difference is then divided by h.
The limit notation “lim h->0” represents the idea of letting h approach zero. It signifies that we are interested in understanding the behavior of the function as the small increment h becomes infinitesimally small. By taking the limit as h approaches zero, we obtain the derivative of the function at the specific point a.
This alternative form of the derivative definition is often referred to as the difference quotient. It provides a way to estimate the instantaneous rate of change of a function at a specific point by considering the average rate of change over a small interval around that point. As h approaches zero, the average rate of change becomes the instantaneous rate of change, which is precisely the derivative.
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