Summarize this article:
113 LearnersLast updated on 6 September 2025
Derivative of ln(1+x)
The derivative of ln(1+x), which is 1/(1+x), serves as a tool to understand how the natural logarithm function changes with respect to a small change in x. Derivatives are crucial for calculating rates of change in various fields. Let us explore the derivative of ln(1+x) in detail.
What is the Derivative of ln(1+x)?
We now understand the derivative of ln(1+x).
It is commonly represented as d/dx (ln(1+x)) or (ln(1+x))', and its value is 1/(1+x).
The function ln(1+x) has a clearly defined derivative, showing it is differentiable within its domain.
The key concepts are mentioned below:
Natural Logarithm Function: ln(1+x).
Chain Rule: Rule for differentiating composite functions like ln(1+x).
Reciprocal: The function that results in 1 when multiplied by the original function.
Derivative of ln(1+x) Formula
The derivative of ln(1+x) can be denoted as d/dx (ln(1+x)) or (ln(1+x))'. The formula used to differentiate ln(1+x) is: d/dx (ln(1+x)) = 1/(1+x) The formula applies for all x where 1+x>0.
Proofs of the Derivative of ln(1+x)
We can derive the derivative of ln(1+x) using proofs.
To demonstrate this, we utilize properties of logarithms and rules of differentiation. Some methods to prove this include:
By First Principle
Using Chain Rule
We will demonstrate how to derive the derivative of ln(1+x) using these methods:
By First Principle
The derivative of ln(1+x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.
To find the derivative of ln(1+x) using the first principle, consider f(x) = ln(1+x).
Its derivative can be expressed as the following limit: f'(x) = limₕ→₀ [f(x+h) - f(x)] / h … (1)
Given f(x) = ln(1+x), we write f(x+h) = ln(1+(x+h)).
Substituting these into equation (1), f'(x) = limₕ→₀ [ln(1+x+h) - ln(1+x)] / h = limₕ→₀ ln[(1+x+h)/(1+x)] / h Using the property ln(a/b) = ln(a) - ln(b), = limₕ→₀ ln[1 + h/(1+x)] / h
Using the limit property, limₕ→₀ ln(1+y)/y = 1 as y→0, f'(x) = 1/(1+x)
Hence, proved.
Using Chain Rule
To demonstrate the differentiation of ln(1+x) using the chain rule, Consider u = 1+x.
Then, ln(1+x) = ln(u). Using the chain rule formula: d/dx [ln(u)] = 1/u · du/dx
Since u = 1+x, du/dx = 1.
Substitute into the chain rule formula: d/dx (ln(1+x)) = 1/(1+x) · 1 = 1/(1+x)
Thus, the derivative is 1/(1+x).
Higher-Order Derivatives of ln(1+x)
When a function is differentiated multiple times, the derivatives obtained are called higher-order derivatives.
Higher-order derivatives can be complex, but they provide deeper insights.
For instance, think of a car's acceleration (second derivative) and how that acceleration changes (third derivative).
For the first derivative of a function, we write f′(x), showing how the function changes or its slope at a certain point.
The second derivative, f′′(x), is derived from the first derivative, and this pattern continues for higher orders.
For the nth derivative of ln(1+x), we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us about the change in the rate of change, continuing this pattern for higher-order derivatives.
Special Cases:
When x = -1, the derivative is undefined because ln(1+x) is undefined at that point. When x = 0, the derivative of ln(1+x) = 1/(1+0), which is 1.
Common Mistakes and How to Avoid Them in Derivatives of ln(1+x)
Students frequently make mistakes when differentiating ln(1+x). These mistakes can be resolved by understanding the solutions. Here are a few common mistakes and ways to solve them:
Mistake 1
Not simplifying the equation
Students may forget to simplify the equation, which can lead to incomplete or incorrect results. They often skip steps and directly arrive at the result, especially when using the chain rule. Ensure that each step is written in order. It might seem tedious, but it is important to avoid errors in the process.
Mistake 2
Ignoring the Domain of ln(1+x)
Students may forget that ln(1+x) is only defined for x>-1. Keep in mind the domain of the function you are differentiating to understand where the function is continuous.
Mistake 3
Incorrect use of Chain Rule
While differentiating functions like ln(1+x), students misapply the chain rule.
For example: Incorrect differentiation: d/dx (ln(1+x²)) = 1/(1+x²).
Applying the chain rule, d/dx (ln(1+x²)) = 1/(1+x²) · 2x = 2x/(1+x²). To avoid this mistake, write the chain rule correctly. Always check for errors in calculation and ensure it is properly applied.
Mistake 4
Not considering Higher-Order Derivatives
Students might not consider higher-order derivatives, which could provide additional insights into the function's behavior. Make sure to differentiate beyond the first derivative when needed, especially in advanced applications.
Mistake 5
Not Applying the Chain Rule Correctly
Students often forget to use the chain rule correctly. This happens when the derivative of the inner function is not considered.
For example: Incorrect: d/dx(ln(2x+1)) = 1/(2x+1).
To fix this error, students should divide the functions into inner and outer parts. Then, ensure each function is differentiated.
For example, d/dx(ln(2x+1)) = 1/(2x+1) · 2 = 2/(2x+1).
Examples Using the Derivative of ln(1+x)
Problem 1
Calculate the derivative of (ln(1+x)·(1+x)²)
Here, we have f(x) = ln(1+x)·(1+x)².
Using the product rule, f'(x) = u′v + uv′
In the given equation, u = ln(1+x) and v = (1+x)².
Let’s differentiate each term, u′ = d/dx (ln(1+x)) = 1/(1+x) v′ = d/dx ((1+x)²) = 2(1+x)
Substituting into the given equation, f'(x) = (1/(1+x)) · (1+x)² + ln(1+x) · 2(1+x)
Let’s simplify terms to get the final answer, f'(x) = (1+x) + 2(1+x)ln(1+x)
Thus, the derivative of the specified function is (1+x) + 2(1+x)ln(1+x).
Explanation
We find the derivative of the given function by dividing the function into two parts. The first step is finding its derivative and then combining them using the product rule to get the final result.
Problem 2
A company records its growth with the function y = ln(1+x) where y represents growth percentage at time x. If x = 0.5 years, calculate the growth rate at this time.
We have y = ln(1+x) (growth function)...(1)
Now, we will differentiate equation (1)
Take the derivative ln(1+x): dy/dx = 1/(1+x)
Given x = 0.5 (substitute this into the derivative) dy/dx = 1/(1+0.5) = 1/1.5 = 2/3
Hence, the growth rate at x=0.5 years is 2/3.
Explanation
We find the growth rate at x=0.5 years as 2/3, which indicates that at this point, the growth is increasing at a rate of 2/3 units per unit time.
Problem 3
Derive the second derivative of the function y = ln(1+x).
The first step is to find the first derivative, dy/dx = 1/(1+x)...(1)
Now we differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [1/(1+x)] = -1/(1+x)²
Therefore, the second derivative of the function y = ln(1+x) is -1/(1+x)².
Explanation
We use the step-by-step process, where we start with the first derivative and then apply the necessary rules to differentiate it again. We simplify to find the second derivative.
Problem 4
Prove: d/dx ((ln(1+x))²) = 2 ln(1+x)/(1+x).
Let’s start using the chain rule: Consider y = (ln(1+x))²
To differentiate, we use the chain rule: dy/dx = 2 ln(1+x) · d/dx [ln(1+x)]
Since the derivative of ln(1+x) is 1/(1+x), dy/dx = 2 ln(1+x) · 1/(1+x)
Substituting y = (ln(1+x))², d/dx ((ln(1+x))²) = 2 ln(1+x)/(1+x) Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. We then replaced ln(1+x) with its derivative. As a final step, we simplified to derive the equation.
Problem 5
Solve: d/dx (ln(1+x)/x)
To differentiate the function, we use the quotient rule: d/dx (ln(1+x)/x) = (d/dx (ln(1+x))·x - ln(1+x)·d/dx(x))/x²
We will substitute d/dx (ln(1+x)) = 1/(1+x) and d/dx (x) = 1 = (1/(1+x)·x - ln(1+x)·1)/x² = (x/(1+x) - ln(1+x))/x² = (x - (1+x)ln(1+x))/(x²(1+x))
Therefore, d/dx (ln(1+x)/x) = (x - (1+x)ln(1+x))/(x²(1+x))
Explanation
In this process, we differentiate the given function using the quotient rule. Then, we simplify the equation to obtain the final result.
FAQs on the Derivative of ln(1+x)
1.Find the derivative of ln(1+x).
2.Can we use the derivative of ln(1+x) in real life?
3.Is it possible to take the derivative of ln(1+x) at the point where x = -1?
4.What rule is used to differentiate ln(1+x)/x?
5.Are the derivatives of ln(1+x) and ln(x) the same?
Important Glossaries for the Derivative of ln(1+x)
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Natural Logarithm: The natural logarithm of a number is its logarithm to the base of the constant e.
- Chain Rule: A rule for differentiating compositions of functions.
- Quotient Rule: A method for differentiating functions that are divided by each other.
- Undefined: A term used to describe a value or expression that does not have meaning in a given context.
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
