Derivative of ln(e^x)

Learn what is the derivative of ln(e^x) with formula. Also understand how to verify the derivative of a function ln(e^x) by using first principle.

Alan Walker-

Published on 2023-06-22

Introduction to the derivative of ln(e^x)

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 differentiation of ln e by using different approaches.

What is the derivative of ln e^x?

The derivative of lne^x is equal to 1 and is represented by d/dx(ln e^x). The ln e^x differentiation with respect to x is an important concept in calculus. This derivative describes the rate of change of the combination of two functions, natural log and exponential functions. It is used extensively in various mathematical and scientific fields.

Additionally, the expression e raised to the power of ln e^x is equal to 1, which highlights the significance of the derivative in understanding the logarithmic properties of x. Therefore, understanding the differentiation of ln e^x is essential for mastering calculus and related subjects. Mathematically the relation between e and ln u is expressed as;

$e^{\ln x} = x$

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

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

Derivative of ln(e) formula

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

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

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

How do you find the derivative of y=lne^x?

There are different ways or methods to derive derivatives of ln e. 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 e^x derivative by using;

  1. First Principle
  2. Implicit Differentiation
  3. Chain rule

Derivative of ln(e) by first principle

The derivative first principle tells that the ln e^x differentiation is equal to the 1. 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 the derivative formula is represented as:

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

You can also use our derivative by definition calculator to easily calculate the derivative of log functions. 

Proof of derivative of ln(e) by first principle

To prove the derivative of ln e by using the first principle, we start by replacing f(x) with ln e, and to differentiate ln3x, replace f(x) with ln(3x).

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

By logarithmic properties,

$f'(u) \;=\; \lim_{h \to 0}\frac{\ln(e^x.e^h)-\ln(e^x)}{h}$


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


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


$f'(u) \;=\; \lim_{t \to 0}\frac{h}{h}$

Therefore, the derivative of ln e^x is;

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

This means that the slope of the curved line at any point on the graph of ln u is equal to the reciprocal of u.

Differentiation of ln e^x 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 ln e differentiation by implicit differentiation

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

$y \;=\; \ln(e^x)$

We can write it as:

$e^y \;=\; e^x$

And, cancelling out the exponential function.

$y \;=\; x$

Applying derivative on both sides,

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

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

Derivative of ln e using chain rule

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

$\frac{dy}{dx} \;=\; \frac{dy}{du}\times \frac{du}{dx}$

Since the function ln e^x is a combination of two functions, therefore we can use the chain rule to prove its derivative.

ln e differentiation proof by chain rule

To find the logarithmic differentiation of the function ln(e^x), we start by assuming that,




Then, the above equation becomes,


Now differentiate y with respect to u and u with respect to x, 


Where the derivative of e^x is equal to e^x.


Where, the derivative ln u is 1/u.

Now using the chain rule formula. 

$\frac{dy}{dx} \;=\; \frac{dy}{du}\times \frac{du}{dx}$

Using the values of derivatives in this formula, 

$\frac{dy}{dx} \;=\; \frac{1}{u}\times e^x$

Substituting u=e^x, 

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

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

Hence the ln e^x differentiation is equal to 1. You can also try a chain rule calculator with steps to differentiate ln e easily.

How to find the derivative of ln(ex) with a calculator?

The easiest way to calculate the ln ex 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(e^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 ex.
  2. Now, select the variable by which you want to differentiate ln ex. Here you have to choose x.
  3. Select how many times you want to calculate derivative of ln(e^x). 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 differentiation of ln e^x within a few seconds.

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

Frequently Asked Questions

What's the derivative of ln e?

The ln e derivative is equal to 1 which is denoted by d/dx(ln e^x)

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

What is the derivative of ln x?

The derivative of ln x with respect to x is equal to 1/x. The formula to calculate ln x derivative is denoted by d/dx(ln x) which is written as:

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

What is the derivative of xln x?

The derivative of xln x is equal to the sum of ln x and 1. It is denoted as d/dx(xlnx). Mathematically, it is expressed as:

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

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