Derivative of ln^3(x)

Learn how to calculate the derivative of ln cube x logarithmic functions with formula. Also understand how to prove the derivative of ln^3(x) by first principle.

Alan Walker-

Published on 2023-06-26

Introduction to the derivative of ln^3(x)

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of ln^3(x) can be calculated by following the rules of differentiation. Or, we can directly find the derivative of ln^3(x) by applying the first principle of differentiation. In this article, you will learn what the derivative of ln^3(x) is and how to calculate the derivative of ln^3(x) by using different approaches. 

What is the derivative of (ln(x))^3?

The derivative of ln^3(x) is equal to 3(ln(x))^2/ x. It is denoted by d/dx [ln^3(x) ]. It is the rate of change of the natural logarithmic function ln cubed x. It is written as;

ln^3(x)=loge3 x

It represents the cubed logarithm of x with base e. 

Derivative of ln^3(x) formula

The formula of derivative of ln3(x) is equal to the reciprocal of 3ln^2 x and x, that is;

d/dx(ln3 (x)) =3ln2(x)/x

How do you prove the derivati6ve of ln^3(x)?

 There are multiple ways to derive derivatives of ln3(x). Therefore, we can prove the derivative of ln^3(x) by using;

  1. Implicit Differentiation
  2. Product Rule

Derivative of ln3(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^3(x) by implicit differentiation.

Proof of derivative of ln3(x) by implicit differentiation 

 To prove the derivative of natural log, we can write it as,

y= ln3(x)

It can be written as;

y1/3 = ln(x)

Converting in exponential form, 

e3√y = x

Applying derivative on both sides,


e3√y/3y1/3 .dy/dx = 1


dy/dx = 3y1/3/ e3√y

Since x = e3√y and y1/3 = ln(x)


dy/dx = 3ln2(x)/x

Use the implicit differentiation calculator to solve the derivative easily.

Derivative of ln cube x using product rule

Another method to find the differentiation of ln^3(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 differentiation of ln^3(x) by product rule 

The function ln x can be written as;

f(x)= ln(x). ln2(x)

Applying derivative with respect to x,

f’(x)=(ln(x). ln2(x))’

By using the product rule calculator with steps

f’(x)=ln(x).(ln2(x))’+ln2(x) (1/x)

f’(x)= 2ln2(x)/x+ln2(x)/x


f’(x) = 3ln2(x)/x

Hence the derivative of ln3(x) is always equal to the ratio of 3ln^2(x) and x.

How to find the derivative of ln^3(x) with a calculator?

The easiest way to calculate the differentation of ln^3(x) is by using an online tool. You can use our differential 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 ln^3(x).
  2. Now, select the variable by which you want to differentiate ln^3(x). Here you have to choose ‘x’.
  3. Select how many times you want to differentiate ln(x) cubed. 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 ln3(x) within a few seconds.


What is the derivative of ln 2 x?

The derivative of ln 2 x is equal to 1/x. We can use the chain rule of differentiation to calculate this derivative, or we can evaluate the derivative of ln2x.

What is the derivative of ln^3(x)?

The derivative of ln^3(x) can be calculated as;

d/dx (ln^3(x)) = 3ln2(x)/x

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