Understanding the Derivative of ex at x 1: A Comprehensive Guide

Introduction

The exponential function, denoted as ex, has a unique property in calculus: its derivative is identical to the function itself. This means that (frac{d}{dx} e^x e^x), making it a cornerstone of differential calculus. In this article, we will explore the value of the derivative of ex at x 1, providing a detailed explanation and proof for its correctness.

Derivative of ex Using First Principles

To understand the derivative of ex, we will employ the concept of first principles. The derivative of a function can be defined using the limit of the difference quotient:

[frac{dy}{dx} limlimits_{h to 0} left{frac{f(x h) - f(x)}{h}right}]

In the case of ex, where (f(x) e^x), the derivative can be calculated as follows:

[frac{d}{dx} e^x limlimits_{h to 0} left{frac{e^{x h} - e^x}{h}right}]

Let's break this down step-by-step:

Substitute (f(x h) e^{x h}) and (f(x) e^x):

[limlimits_{h to 0} left{frac{e^{x h} - e^x}{h}right}]

Factor out (e^x) from the numerator:

[limlimits_{h to 0} left{e^x cdot frac{e^h - 1}{h}right}]

Rewrite the expression as a product of two limits:

[e^x cdot limlimits_{h to 0} left{frac{e^h - 1}{h}right}]

Note that (limlimits_{h to 0} left{frac{e^h - 1}{h}right} 1), which is a known limit in calculus.

Therefore:

[frac{d}{dx} e^x e^x cdot 1 e^x]

This derivation confirms that the derivative of ex is ex itself.

Evaluating the Derivative at x 1

Given the derivative of ex is ex, we can directly evaluate the derivative at any point. Specifically, for x 1:

[left(frac{d}{dx} e^xright) text{at } x 1 e^x text{ at } x 1 ]

(left(frac{d}{dx} e^xright) text{ at } x 1 e^1 )

(left(frac{d}{dx} e^xright) text{ at } x 1 e)

The value of e at x 1 is simply e, making it a straightforward calculation.

Conclusion

The derivative of the exponential function ex at any point, including x 1, is a fundamental concept in calculus. By understanding the derivation using first principles, we can confidently state that the derivative of ex at x 1 is e. This property makes the exponential function unique and essential in various mathematical and scientific applications.