Understanding the Limit Definition of Derivative | A Mathematical Approach to Calculating the Instantaneous Rate of Change

Limit Definition of Derivative

The limit definition of derivative establishes the exact mathematical definition of how to find the derivative of a function at a specific point

The limit definition of derivative establishes the exact mathematical definition of how to find the derivative of a function at a specific point. It allows us to calculate the instantaneous rate of change or the slope of a curve at a particular point.

Let’s consider a function f(x), and we want to find the derivative of this function at a point x=a. The limit definition of derivative is given as follows:

f'(a) = lim(h→0) [f(a+h) – f(a)] / h

This equation represents the derivative of f(x) at x=a, denoted as f'(a). The term “lim(h→0)” indicates that we are taking the limit as the change in x, represented by h, approaches zero. By computing the difference quotient, which is [f(a+h) – f(a)] / h, and taking the limit as h approaches zero, we obtain the derivative at that point.

In simpler terms, this formula tells us to find the slope between two points on the graph of the function that are collectively approaching the point (x=a). The two points are (a+h, f(a+h)) and (a, f(a)), and the slope of the line connecting them is calculated as the difference in y-coordinates divided by the difference in x-coordinates.

The limit definition of derivative is fundamental because it serves as the basis for deriving various rules and formulas used to find derivatives of different types of functions. Once we find the derivative at a specific point, we can determine the slope of the tangent line to the graph of the function at that point, which provides valuable information in fields such as physics, engineering, and economics.

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