What is the Derivative of ln(x2)?

Introduction

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

What is the derivative of ln(x2)?

The derivative of ln x with respect to 'x' is 1/x. Denoted as d/dx [ln(x^2)], it represents the rate of change of the natural logarithmic function ln x. In other words, it quantifies how the value of ln x^2 changes as x varies. Alternatively, ln(x^2) can be expressed as;

ln(x2)=loge x2

which denotes the logarithm of 2x with base e. Understanding the derivative of ln(x^2) is essential for understanding different concepts of calculus and its practical applications.

Derivative of ln(x^2) formula

The derivative formula for ln(x^2) is given as 1/x. It represents the reciprocal of x. In mathematical notation, we can express it as:

d/dx(ln(x2)) =1/x

This derivative equation enables us to calculate the rate of change of ln(x^2) with respect to x. Understanding and applying this formula is fundamental in various mathematical and scientific disciplines.

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

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

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

Derivative of ln(x2) by first principle

According to the first principle of derivative the ln(x2) 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 allows us to determine the rate of change of a function at a specific point by using the derivative definition calculator. 

Proof of derivative of ln(x2) by first principle

To prove the derivative of ln(x2) by using first principle, we start by replacing f(x) by ln x. 

f’(x)=lim{ln(x+h)2-ln(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=hx 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(x2) is:

f’(x)=2/x

This method can be also used to calculate the derivative of ln x.

Derivative of ln(x2) using implicit differentiation 

Since in implicit differentiation, we differentiate a function with two variables. Here we will prove the derivative of ln(x2) by implicit differentiation.

Proof of derivative of ln(x2) by implicit differentiation 

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

y=ln(x2)

Converting in exponential form, 

ey = x2

Applying derivative on both sides,

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

ey.dy/dx = 2x

Now,

dy/dx=2x/ey

Since x2 = ey

Therefore, 

dy/dx = 2/x

Derivative of ln(x2) 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;

d/dx{f(x)g(x)}=f(x)g’(x)+g(x)f’(x)

The derivative of ln x square can be calculated by using the product rule formula because the function ln x^2 can be written as the combination of two functions. 

Proof of derivative of ln(x2) by product rule 

The function ln x can be written as;

f(x)= 1. ln(x2)

Applying derivative with respect to x,

f’(x)=(1. ln(x2))’

Applying product rule, 

f’(x)=1.(ln(x2))’+ln(x2) (0)

f’(x)=1.(2/x)+0

Therefore,

f’(x) = 2/x

Hence the derivative of ln(x2) is always equal to the reciprocal of x. Use our product rule calculator to verify above manual calculations. 

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

The most easy method to calculate the derivative of ln(x^2) is by using an online tool. Our derivative calculator can assist you in this process, and here's a step-by-step guide on how to use it effectively.

1. Enter the function as ln(x) in the "enter function" box. Since we want to calculate the derivative of ln(x^2), we consider it as ln(x) in the initial step.
2. Specify the variable by which you intend to differentiate ln(x^2). In this case, select 'x' as the differentiating variable.
3. Determine the number of times you want to differentiate ln(x^2). For instance, if you wish to find the second derivative, choose 2. You can select higher values for higher-order derivatives.
4. Click on the calculate button to initiate the computation. Within a few seconds, the derivative of ln(x^2) will be displayed.

By following these steps and using the derivative calculator with steps, you can obtain the derivative of ln(x^2) conveniently and efficiently

FAQ’s

What is the derivative of exponential and derivative of logarithmic functions? 

The derivative of an exponential function always results in an exponential function. Whereas the derivative of a logarithmic function results in the reciprocal of the variable involved.

What is the derivative of ln(x2)?

The derivative of ln(x2) can be calculated as;

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