What is the Formula of Derivative?

The general formula for the derivative of a function, often referred to as the "definition of the derivative," is:

y' = lim Δx→0 Δy/Δx

This can also be expressed as:

dy/dx = lim Δx→0 (f(x + Δx) - f(x)) / Δx

Explanation of the Formula

This formula defines the derivative as the limit of the difference quotient as Δx (the change in x) approaches zero. Let's break down each part:

• y' (or dy/dx): This represents the derivative of the function y with respect to x. It signifies the instantaneous rate of change of y with respect to x.
• lim Δx→0: This is the limit as Δx approaches 0. The concept of a limit is crucial because we want to examine the rate of change at a single point, not over a finite interval.
• Δy/Δx: This is the difference quotient, which represents the average rate of change of the function over the interval from x to x + Δx. Δy represents the change in y, and Δx represents the change in x. Specifically, Δy = f(x + Δx) - f(x).
• f(x): Represents the original function whose derivative is to be found.
• f(x + Δx): Represents the value of the function at the point x + Δx.

How to Interpret the Formula

The derivative, dy/dx, gives the slope of the tangent line to the curve of the function f(x) at a particular point x. By finding the limit of the difference quotient as Δx approaches zero, we essentially zoom in on the function at a single point, allowing us to determine the slope of the curve at that precise location.

Example

Let's consider finding the derivative of f(x) = x² using this formula:

1. Set up the difference quotient:

   (f(x + Δx) - f(x)) / Δx = ((x + Δx)² - x²) / Δx

2. Expand and simplify:

   ((x² + 2xΔx + (Δx)²) - x²) / Δx = (2xΔx + (Δx)²) / Δx = 2x + Δx

3. Take the limit as Δx approaches 0:

   lim Δx→0 (2x + Δx) = 2x

Therefore, the derivative of f(x) = x² is 2x.

Summary

The formula for the derivative allows us to determine the instantaneous rate of change of a function at any given point. It is a fundamental concept in calculus and has numerous applications in various fields, including physics, engineering, and economics.