Introduction to the derivative of ln(x+1)

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

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

The derivative of ln(x+1) with respect to the variable 'x' is equal to 1/(x+1). This derivative is denoted as d/dx [ln(x+1)]. It represents the rate of change of the natural logarithmic function ln(x). The expression ln(x+1) can also be written as loge(x+1), indicating that it represents the logarithm of (x+1) with base e.

ln(x+1)=log_e(x+1)

Derivative of ln(x+1) formula

The derivative of ln(x+1) with respect to the variable 'x' is equal to the reciprocal of (x+1), which can be expressed as 1/(x+1). This derivative is denoted as d/dx [ln(x+1)]. It represents the rate of change of the natural logarithmic function ln(x) such as,

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

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

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

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

Derivative of ln(x+1) by first principle

According to the first principle of derivative, the ln(x+1) derivative is equal to 1/x+1. The derivative of a function by the 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 definition of the derivative calculator.

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

To prove the derivative of ln(x+1) by using first principle, we start by replacing f(x) by ln x+1. 

f’(x)=lim{ln2(x+1+h)-ln(x+1)/h}

By logarithmic properties,

f’(x)=lim {ln(x+1+h/x+1)/h}

Simplifying,

f’(x)=lim {ln(1+h/x+1)/h}

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

f’(x)=lim {ln(1+t)/(x+1)t}

And, 

f’(x)=lim ln (1/(x+1)t) ln (1+t) 

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

f’(x)=(1/x+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(x+1) is;

f’(x)=1/x+1

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

Derivative of ln(x+1) using implicit differentiation 

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

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

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

y=ln(x+1)

Converting in exponential form, 

ey = x+1

Applying derivative on both sides,

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

ey.dy/dx = 1

Now,

dy/dx=1/ey

Since x+1=ey

Therefore, 

dy/dx=1/x+1

Hence we have verified the derivative of ln(x+1) by using the implicit differentiation. The implicit function theorem can also be used to calculate the rate of a change of an implicit function. 

Derivative of ln(x+1) using product rule

Another method to find the differentiation of ln(x+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.

Proof of derivative of ln(x+1) by product rule 

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

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

Applying derivative with respect to x,

f’(x)=(1. ln(x+1))’

Applying product rule, 

f’(x)=1.(ln(x+1))’+ln(x+1) (0)

f’(x)=1.(1/x+1)+0

Therefore,

f’(x)=1/x+1

Hence the derivative of ln(x+1) is always equal to the reciprocal of x. Also, use our porduct rule calculator with steps to do calculations of derivative of f(x)g(x).

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

The easiest way to calculate the derivative of ln(x+1) 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(x+1).
2. Now, select the variable by which you want to differentiate ln(x+1). Here you have to choose ‘x’.
3. Select how many times you want to differentiate ln(x+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(x+1) within a few seconds.

FAQ’s

How do you take the derivative of ln x+1?

To find the derivative of ln(x+1), you need to differentiate it with respect to x. The first derivative of ln(x+1) is;

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

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

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

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