Derivative of ln^2(x)

Learn the easiest way to prove the derivative of ln^2(x) with formula and calculations. Also understand that (lnx)^2 is the same as ln^2x.

Alan Walker-

Published on 2023-06-26


Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of ln^2(x) can be calculated by following the rules of differentiation. Or, we can directly find the derivative of ln^2(x) by applying the first principle of differentiation. In this article, you will learn what the derivative of ln^2(x) is and how to calculate the derivative of ln^2(x) by using different approaches. 

What is the derivative of ln^2(x)?

The derivative of ln^2x is equal to 2ln x/ x. It is denoted by d/dx [ln2(x)]. It is the rate of change of the natural logarithmic function ln squared x. It is written as;

Ln2(x)=loge2 x

It represents the squared logarithm of x with base e. 

Derivative of ln2(x) formula

The formula of derivative of ln2(x) is equal to the ratio between 2ln x and x, that is;

d/dx(ln2(x)) =2ln(x)/x

It can be calculated by using different differentiation rules. Let’s understand how to prove the derivative of (ln(x))^2.

How do you prove the derivative of ln2(x)?

 There are multiple ways to derive derivative of ln²x. Therefore, we can prove the derivative of ln2(x) by using;

  1. Implicit Differentiation
  2. Product Rule

Derivative of ln2(x) using implicit differentiation 

Implicit differentiation is a technique of solving derivatives of implicit functions. It plays an important role in differentiating logarithmic functions. Since the ln square x can be treated as an implicit function, we can use this method for logarithmic differentiation. Here we will prove the derivative of ln squared x by implicit differentiation.

Proof of ln^2(x) derivative by implicit differentiation 

 To prove the derivative of natural log by using implicit differentiation, we can write it as,


It can be written as;

√y = ln(x)

Converting in exponential form, 

e√y = x

Applying derivative on both sides,


e√y/2√y .dy/dx = 1


dy/dx = 2√y/ e√y

Since x = e√y and √y = ln(x)



You can also use our implicit derivative calculator to find the derivative of ln squared x easily.

Derivative of ln²x using product rule

Another method to find the differentiation of ln^2(x) is the product rule formula which is used in calculus to calculate the derivative of the product of two functions. Specifically, the product rule is used when you need to differentiate two functions that are multiplied together. The product rule formula for a product of two functions is:


In this formula, u and v are functions of x, and du/dx and dv/dx are their respective derivatives with respect to x.

Proof of derivative of ln^2(x) by product rule 

The function ln x can be written as;

f(x)= ln(x). ln(x)

Applying derivative with respect to x,

f’(x)=(ln(x). ln(x))’

Applying product rule, 

f’(x)=ln(x).(ln(x))’+ln(x) (1/x)

f’(x)= ln(x)/x+ln(x)/x


f’(x) = 2ln(x)/x

Hence the derivative of ln^2(x) is always equal to the ratio of 2ln x and x. Also, use the product rule calculator to differentiate ln squared x. 

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

The easiest way to calculate the ln^2(x) derivative is by using an online tool. You can use our dy/dx 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 ln²x.
  2. Now, select the variable by which you want to differentiate ln2(x). Here you have to choose ‘x’.
  3. Select how many times you want to differentiate ln2(x). In this step, you can choose 2 for second, 3 for third derivative and so on.
  4. Click on the calculate button. After this step, you will get the derivative of ln2(x) within a few seconds.


Is (lnx)^2 equivalent to ln^2 x?

Yes (lnx)2 is the same as ln2 (x). It is because (lnx)2 can be written as the product of lnx two times. It can also be written as 2lnx as in logarithmic rules, the power in log can be shifted to the left side as a multiple. 

What is the derivative of ln^2(x)?

The derivative of ln^2(x) can be calculated as;

d/dx (ln^2(x)) = 2ln(x)/x

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