# Derivative of ln(2x^2)

Learn how to calculate the derivative of ln(2x^2) logarithmic function with formula. Also understand how to prove the derivative of ln 2x^2 by first principle.

Alan Walker-

Published on 2023-06-26

## Introduction to the derivative of ln(2x^2)

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of ln(2x^2) can be calculated by following the rules of differentiation. Or, we can directly find the ln(2x^2) derivative by applying the first principle of differentiation. In this article, you will learn what the derivative of ln 2x squared is and how to calculate the derivative of ln(2x2) by using different approaches.

## What is the derivative of ln2x^2?

The derivative of ln 2x^2 is equal to 2/ x. It is denoted by d/dx [ln(2x2)]. It is the rate of change of the natural logarithmic function ln 2x squared.

Mathematically, the funciton ln(2x^2) is written as;

ln(2x^2)=loge 2x^2

It represents the logarithm of 2x^2 with base e.

## Derivative of ln(2x^2) formula

The formula of derivative of ln 2x^2 is equal to the fraction of 2 and x, that is;

d/dx(ln(2x2)) =2/x

## How do you prove the derivative of ln(2x2)?

There are multiple derivative rules to derive derivatives of ln(2x2). Therefore, we can prove the derivative of ln(2x2) by using;

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

## Derivative of ln2x^2 by first principle

The derivative first principle says that the ln(2x^2) derivative is equal to the reciprocal of x. The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,

f’(x)=f(x+h)-f(x)/h

This formula is used in a derivative by definition calculator which provides you a stepwise easy solution. Let’s understand how to verify the derivative of ln 2 x 2 by using the first principle.

## Proof of derivative of ln(2x2) by first principle

To prove the ln(2x^2) derivative by using first principle, we start by replacing f(x) by ln x.

f’(x)=lim{ln2(x+h)2-ln2(x2)/h}

By logarithmic properties,

f’(x)=lim {ln(x+h/x)2/h}

Simplifying,

f’(x)=lim {2ln(1+h/x)/h}

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

f’(x)=lim 2{ln(1+t)/xt}

And,

f’(x)=lim 2ln (1/xt) ln (1+t)

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

f’(x)=(2/x) lim ln(1+t)/t

Hence by limit formula, we know that,

lim ln(1+t)/t =ln e =1

Therefore, the derivative of ln(2x2) is:

f’(x)=2/x

## Derivative of ln2x^2 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 2x squared can be treated as an implicit function, we can use this method for logarithmic differentiation. Here we will prove the derivative of ln(2x)^2 by implicit differentiation.

## Proof of derivative of ln(2x2) by implicit differentiation

To differentiate ln2x^2, we can write it as,

y=ln(2x2)

Converting in exponential form,

ey = 2x2

Applying derivative on both sides,

d/dx(ey)=d/dx(2x2)

ey.dy/dx = 4x

Now,

dy/dx=4x/ey

Since x2 = ey

Therefore,

dy/dx = 2/x

## Derivative of ln(2x^2) using product rule

Another method to find the differentiation of ln(2x^2) 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:

d/dx{f(x)g(x)}=f(x)g’(x)+g(x)f’(x)

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(2x^2) by product rule

The function ln 2x^2 can be written as;

f(x)= 1. ln(2x2)

Applying derivative with respect to x,

f’(x)=(1. ln(2x2))’

Using the product rule calculator

f’(x)=1.(ln(2x2))’+ln(2x2) (0)

f’(x)=1.(2/x)+0

Therefore,

f’(x) = 2/x

Hence the derivative of ln(2x2) is always equal to the ratio of 2 and x.

## How to find the derivative of ln(2x2) with a calculator?

The easiest way to calculate the derivative of ln(2x2) is by using an online tool. 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 ln(2x2).
2. Now, select the variable by which you want to differentiate ln2x^2. Here you have to choose ‘x’.
3. Select how many times you want to differentiate ln(2x^2). 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 ln(2x2) within a few seconds.