Introduction to the derivative of ln(2x+1)

Derivatives have a wide range of applications in almost every field of engineering and science. The ln(2x+1) derivative can be calculated by following the rules of differentiation. Or, we can directly find the differentiation of ln(2x+1) by applying the first principle of differentiation. In this article, you will learn what the differentiation of ln(2x+1) is and how to calculate the derivative of ln(2x+1) by using different approaches. 

What is the derivative of ln(2x+1)?

The derivative of ln(2x+1) with respect to the variable 'x' is equal to 2/(2x+1). This derivative is denoted as d/dx [ln(2x+1)]. It represents the rate of change of the natural logarithmic function ln(x). The expression ln(2x+1) can also be written as;

 ln(2x+1)=log_e(2x+1)

which indicates that it represents the logarithm of (2x+1) with base e. The process of finding derivative of ln2x1 is known as logarithmic differentiation.

Derivative of ln2x+1 formula

 The formula for the derivative of ln 2x+1 with respect to the variable 'x' is not equal to the reciprocal of (2x+1). It can be expressed as 1/(2x+1). This derivative is denoted as d/dx [ln(2x+1)]. It represents the rate of change of the natural logarithmic function ln(2x+1) such as:

d/dx(ln(2x+1)) =1/2x+1

How do you prove the differentiation of ln(2x+1)?

 There are multiple differentation rules to derive derivatives of ln(2x+1). Therefore, we can prove the differentiation of ln(2x+1) by using;

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

Derivative of ln(2x+1) by first principle

The derivative first principle says that the ln(2x+1) derivative is equal to 2/(2x+1). 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 limit definition of the derivative.

Proof of derivative of ln(2x+1) by first principle

To prove the derivative of ln2x+1 by using first principle, replace f(x) by ln(2x+1). 

f’(x)=lim{ln(2x+1+h)-ln(2x+1)/h}

Replace f(x) by ln x to calculate the ln x derivative.

By logarithmic properties,

f’(x)=lim {ln(2x+1+h/2x+1)/h}

Simplifying,

f’(x)=lim {ln(1+h/2x+1)/h}

Suppose t=h/2x+1 and h=t(2x+1). When h approaches zero, t will also approach zero.

f’(x)=lim {ln(1+t)/(2x+1)t}

And, 

f’(x)=lim ln (1/(2x+1)t) ln (1+t) 

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

f’(x)=(1/2x+1) lim ln(1+t)1/t

Hence by limit formula, we know that,

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

Therefore, the derivative of ln 2x+1 is;

f’(x)=2/2x+1

Hence we have verified the ln(2x+1) differentiation. You can also use the first derivative calculator to find the first ln(2x+1) derivative.

Differentiation of ln(2x+1) using implicit differentiation 

Since in implicit differentiation, we differentiate a function with two variables. Here we will prove the derivative of ln(2x+1) by implicit differentiation.

Proof of ln(2x+1) derivative by implicit differentiation 

 To prove the derivative of natural log, we can start by writing it as,

y=ln(2x+1)

Converting in exponential form, 

ey = 2x+1

Applying derivative on both sides,

d/dx(ey)=d/dx(2x+1)

ey.dy/dx = 2

Now,

dy/dx=2/ey

Since 2x+1=ey

Therefore, 

dy/dx=2/2x+1

Hence the differentiation of ln(2x+1) can also be calculated by using implicit differentiation. You can use our implicit differentiation calculator to calculate derivative by using an online tool. 

Derivative of ln(2x+1) using product rule

Another method to find the differentiation of ln(2x+1) 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. This formula can also be used to calculate the derivative of ln(x+1).

Proof of differentiation of ln(2x+1) by product rule 

The function ln(2x+1) can be written as;

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

Applying derivative with respect to x,

f’(x)=(1. ln(2x+1))’

Now by using the product rule calculator, 

f’(x)=1.(ln(2x+1))’+ln(2x+1) (0)

f’(x)=1.(2/2x+1)+0

Therefore,

f’(x)=2/2x+1

Hence the derivative of ln(2x+1) is always equal to the ratio of 2 and 2x+1.

How to find the differentiation of ln(2x+1) with a calculator?

The easiest way to calculate the derivative of ln(2x+1) is by using an online tool. You can use our differentation 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+1.
2. Now, select the variable by which you want to differentiate ln(2x+1). Here you have to choose ‘x’.
3. Select how many times you want to differentiate ln2x+1. 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 2x+1 within a few seconds.

FAQ’s

Why natural log is the inverse?

Since logarithmic functions are one-to-one functions and the natural log is a logarithmic function to the base of e. Therefore the inverse of ln x exists and the inverse of natural log is exponential functions.

What is the derivative of ln(2x+1)?

The derivative of ln(2x+1) can be calculated as;

d/dx (ln(2x+1)) = 2/2x+1