formal version of def. of derivative
The formal definition of the derivative of a function f(x) at a point x = c is given by the limit:
lim (h → 0) [f(c + h) – f(c)] / h
This definition represents the rate of change of the function f(x) at the specific point x = c
The formal definition of the derivative of a function f(x) at a point x = c is given by the limit:
lim (h → 0) [f(c + h) – f(c)] / h
This definition represents the rate of change of the function f(x) at the specific point x = c. It calculates the slope of the tangent line to the graph of the function at that point.
In this definition, h represents a small change in the value of x around the point c. As h approaches zero, it signifies an infinitesimally small change in x. By calculating the difference quotient [f(c + h) – f(c)] / h, we obtain the average rate of change between x = c and x = c + h. Taking the limit as h approaches zero provides the instantaneous rate of change, which is the derivative at the point x = c.
This definition generalizes the concept of the derivative for all types of functions and allows us to find the derivative at any point, as long as the limit exists.
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