Formal definition of derivative
The formal definition of a derivative is given as:
The derivative of a function f at a specific point x, denoted as f'(x) or dy/dx, is the limit of the difference quotient as the change in x approaches zero
The formal definition of a derivative is given as:
The derivative of a function f at a specific point x, denoted as f'(x) or dy/dx, is the limit of the difference quotient as the change in x approaches zero.
Mathematically, it is expressed as:
f'(x) = lim (h->0) [(f(x + h) – f(x))/h]
In this definition, h represents a small change or increment in the value of x, and as h approaches zero, it signifies an infinitesimally small change in x. The difference quotient [(f(x + h) – f(x))/h] measures the average rate of change of the function f between x and x+h.
By taking the limit as h approaches zero, we obtain the instantaneous rate of change or slope of the function at a particular point x. This instantaneous rate of change is known as the derivative of the function f at x.
The derivative represents the slope of the tangent line to the graph of the function at a given point. It provides information about the rate of change of the function, whether it is increasing or decreasing, and the concavity of the curve.
The formal definition of the derivative can be used to compute derivatives of various functions using rules such as the power rule, product rule, quotient rule, and chain rule, among others. These rules allow us to differentiate complicated functions and solve problems in various fields such as physics, economics, engineering, and more.
More Answers:
The Intermediate Value Theorem: Understanding the Behavior of Continuous Functions and Proving the Existence of ValuesUnderstanding the Average Rate of Change in Mathematics: Calculation and Interpretation
Discovering the Instantaneous Rate of Change: Calculating Derivatives and Tangent Line Slopes