# Derivative of ln(u)

Learn what is the derivative of ln(u) with formula. Also understand how to verify the derivative of an algebraic function ln(u) by using first principle.

Alan Walker-

Published on 2023-06-21

## Introduction to the derivative of ln

Derivatives have a wide range of applications in almost every field of engineering and science. The ln u derivative can be calculated by following the derivative rules. Or, we can directly find the ln derivative by applying the first principle of differentiation. In this article, you will learn what the derivative ln u is and how to calculate the derivative of ln(u) by using different approaches.

## What is the derivative of ln(u)?

The ln(u) derivative is equal to 1/u and is represented by d/dx(ln x. The derivative of ln u with respect to u is a crucial concept in calculus. This derivative describes the rate of change of the natural logarithmic function ln(x), which is used extensively in various mathematical and scientific fields.

Additionally, the expression e raised to the power of ln u is equal to u, which highlights the significance of the derivative in understanding the logarithmic properties of u. Therefore, understanding the derivative of ln(u) is essential for mastering calculus and related subjects. Mathematically the relation between e and ln u is expressed as;

$e^{\ln u} = u$

It represents the logarithm of u with base e Similarly, the derivative of ln x is:

.$e^{\ln x} = x$

### Derivative of ln formula

The derivative of natural logarithm, ln u, is calculated using the formula,

$\frac{d}{du} (\ln u) = \frac{1}{u}$

This formula can be proven using the limit definition of a derivative. The ln(u) derivative is an essential concept in calculus and finds application in various mathematical and scientific fields. By understanding the derivative of ln u, you can solve derivative problems related to optimization, rates of change, and exponential growth.

## How do you prove the ln u derivative formula?

There are different ways or methods to derive derivatives of ln u. These methods allow you to find the rate of change or slope of the ln u function at any point, which is useful in many applications. We can prove the ln u derivative by using;

1. First Principle
2. Implicit Differentiation
3. Product Rule

## Derivative of ln(u) by first principle

The derivative first principle tells that the differentiation of ln u is equal to the 1/u. The derivative of a function by first principle refers to finding the slope of a curve by using algebra. It is also known as the delta method. Mathematically, the first principle of derivative formula is represented as: