Mathematical Differentiation Meaning In Hindi

The Derivative: A Limit Definition in Hindi

Introduction

The derivative is a fundamental concept in calculus that measures the rate of change of a function. In this blog post, we will explore the formal definition of the derivative as a limit in Hindi. We will also provide a numerical example to illustrate the concept.

Formal Definition

Let $f(x)$ be a function defined on an open interval $(a,b)$. The derivative of $f(x)$ at $x=c$ is defined as:

$f\text{'}(c)\; =\; lim\_\{h->0\}\; [f(c+h)-f(c)]/h$

where $h$ is a small increment.

Numerical Example

Let's consider the function $f(x)\; =\; x^2$. To find the derivative of $f(x)$ at $x=2$, we substitute $x=2$ into the definition of the derivative:

$f\text{'}(2)\; =\; lim\_\{h->0\}\; [f(2+h)-f(2)]/h$ $=\; lim\_\{h->0\}\; [(2+h)^2-2^2]/h$ $=\; lim\_\{h->0\}\; [4+4h+h^2-4]/h$ $=\; lim\_\{h->0\}\; [4h+h^2]/h$ $=\; lim\_\{h->0\}\; 4+h$ $=4$

Therefore, the derivative of $f(x)$ at $x=2$ is 4.