# Derivative of ln x

Find about the derivative of ln x along with how to verify the derivative of lnx by using implicit differentiation and product rule.

Alan Walker-

Published on 2023-05-26

## Introduction to the derivative of ln

Derivatives have a wide range of applications in almost every field of engineering and science. The ln x derivative can be calculated by following the rules of differentiation. 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 x is and how to calculate the derivative of ln(x) by using different approaches.

## What is the derivative of lnx?

The derivative of ln(x) with respect to x is a crucial concept in calculus. It is represented by d/dx(ln x) and has a constant value of 1/x. 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 x is equal to x, which highlights the significance of the derivative in understanding the logarithmic properties of x. Therefore, understanding the derivative of ln(x) is essential for mastering calculus and related subjects. Mathematically the relation between e and ln x is expressed as;

$e^{\ln x} = x$

It represents the logarithm of x with base e.

### Derivative of ln formula

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

$\frac{d}{dx} (\ln x) = \frac{1}{x}$

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

## How do you prove the derivative of ln x?

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

1. First Principle

2. Implicit Differentiation

3. Product Rule

## Derivative of ln(x) by first principle

The derivative first principle tells that the ln x differentiation is equal to the 1/x. 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:

$f'(x) \;=\;\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$

## Proof of derivative of ln x by first principle

To prove the derivative of ln(x) by using first principle, we start with replacing f(x) by ln x and to differentiate ln4x, replace f(x) by ln(4x).

$f'(x) \;=\; \lim_{h \to 0}\frac{ln (x+h) -ln x}{h}$

By logarithmic properties,

$f'(x) \;=\; \lim_{h \to 0}\frac{\frac{ln(x+h)}{x}}{h}$

Simplifying,

$f'(x) \;=\; \lim_{h \to 0}\frac{ln(\frac{1+h}{x})}{h}$

Suppose t = h / x and h=xt. When h approaches zero, t will also approach zero.

$f'(x) \;=\; \lim_{t to 0} \frac{ln (1+t)}{xt}$

And,

$f'(x) \;=\; \lim_{t \to 0}\frac{1}{xt} ln (1+t)$

By logarithmic properties, we can write the above equation as,

$f'(x) \;=\; \frac{1}{x}\lim_{t to 0}\ln (1+t)^{\frac{1}{t}}$

Hence by limit formula, we know that,

$\lim_{t \to 0}\ln (1+t)^{\frac{1}{t}} \;=\; \ln e \;=\; 1$

Therefore, the ln x derivative is;

$f'(x) \;=\; (\frac{1}{x})$

This means that the slope of the curve at any point on the graph of ln x is equal to the reciprocal of x.

## Derivative of lnx using implicit differentiation

Since in implicit differentiation, we differentiate a function with two variables. Here we will prove the ln derivative by implicit differentiation.

## Proof of lnx derivative by implicit differentiation

We can easily differentiate ln x by using the implicit differentiation. Now to prove the derivative of natural log, we can write it as,

$y \;=\; \ln x$

Converting in exponential form,

$e^y \;=\; x$

Applying derivative on both sides,

$\frac{d}{dx}(e^y) \;=\; \frac{d}{dx}(x)$

$e^y.\frac{dy}{dx} \;=\; 1$

Now,

$\frac{dy}{dx} \;=\; \frac{1}{e^y}$

Since

$e^y \;=\; x$

Therefore, the derivative of ln(x) is,

$\frac{dy}{dx} \;=\; \frac{1}{x}$

Use our implicit differentiation calculator to verify the above calculations.

## Differentiation of ln x using product rule

The product rule in derivatives is used when we have to calculate derivatives of two functions at a time. The product rule for two functions says that;

$\frac{d}{dx}(f(x)g(x)) \;=\; f(x)g(x) \;+\; g(x)f (x)$

Since the function ln x can be written as a product of two functions, therefore we can use it to prove its derivative.

## Derivative of ln(x) proof by product rule

To find the derivative ln, the function ln x can be written as;

$f(x) \;=\; 1.ln x$

Applying derivative with respect to x,

$f'(x) \;=\;(1.ln x )'$

Applying product rule,

$f'(x) \;=\;1.x +ln x. (1)$

$f'(x)=1.\frac{1}{x} + 0$

Therefore,

$f'(x) \;=\; \frac{1}{x}$

Hence the derivative of lnn x is always equal to the reciprocal of x. You can also try product rule calculator with steps to differentiate ln x easily.

## How to find the derivative of ln x with a calculator?

The easiest way to calculate the ln derivative is by using an online derivative finder. You can use our derivative calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

1. Write the function as ln x in the enter function box. In this step, you need to provide input value as a function as you have to calculate the derivative of the function ln x.

2. Now, select the variable by which you want to differentiate ln x. Here you have to choose &x.

3. Select how many times you want to calculate derivatives of ln. In this step, you can choose 2 for second, 3 for third order derivative and so on.

4. Click on the calculate button. After this step, you will get the derivative of lnx within a few seconds.

After completing these steps, you will receive the ln x derivative within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.

### Is the derivative of ln always 1/x?

Yes the lnx derivative is always equal to 1/x. Because when we differentiate a function with natural log, it always results in the reciprocal of the function. The derivative ln x is written as;

d / dx (ln x) = 1 / x

### What is the derivative of ln 5x?

The derivative of ln(5x) can be calculated as;

$\frac{d}{dx}(\ln 5x) = \frac{1}{x}$

The derivative of ln5x is also same as the derivative of lnx.

### What is lnx equal to?

The function ln x is a logarithmic function whose base is e which is an exponential function. It is written as;

$\ln x =x$

It represents the logarithm of x with base e.

### Why is it called natural log?

It is said to be natural because e is the universal rate of growth. Therefore, ln could be considered a universal way to find out how long things take to grow. When you see ln x, it means that the amount of time to grow to x.