Understanding Derivatives | The Key to Analyzing Functions and Their Rates of Change

Definition of derivative of function as a function

The derivative of a function can be defined as a new function that describes the rate of change of the original function at each point

The derivative of a function can be defined as a new function that describes the rate of change of the original function at each point. The derivative function, denoted as f'(x) or dy/dx, represents the slope of the tangent line to the graph of the original function at any given point.

Mathematically, given a function f(x), the derivative function f'(x) is found by taking the limit of the difference quotient as the change in x approaches zero:

f'(x) = lim(h->0) [(f(x + h) – f(x))/h]

This definition means that the derivative function measures how the value of the original function changes with respect to a small change in the input variable x. It provides information about the instantaneous rate of change, or the slope of the function, at any specific point.

The derivative function can also be interpreted geometrically as the slope of the tangent line to the graph of the original function. At a particular point (x, f(x)), the derivative f'(x) gives the slope of the line that best approximates the curve of f(x) near that point.

The derivative has numerous applications in mathematics and various fields such as physics, economics, and engineering. It plays a fundamental role in optimization problems, curve sketching, and studying the behavior of functions.

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