Derivative of ln(x+3)
2026-02-28 06:19 Diff
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Last updated on August 5, 2025

We use the derivative of ln(x+3), which is 1/(x+3), as a measuring tool for how the logarithmic function changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now talk about the derivative of ln(x+3) in detail.

What is the Derivative of ln(x+3)?

We now understand the derivative of ln(x+3). It is commonly represented as d/dx (ln(x+3)) or (ln(x+3))', and its value is 1/(x+3). The function ln(x+3) has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Logarithmic Function: ln(x+3) is the natural logarithm of (x+3). Chain Rule: Used when differentiating ln(x+3) since it involves an inner function (x+3). Reciprocal: The derivative involves taking the reciprocal of the inner function.

Derivative of ln(x+3) Formula

The derivative of ln(x+3) can be denoted as d/dx (ln(x+3)) or (ln(x+3))'. The formula we use to differentiate ln(x+3) is: d/dx (ln(x+3)) = 1/(x+3) The formula applies to all x where x+3 > 0.

Proofs of the Derivative of ln(x+3)

We can derive the derivative of ln(x+3) using proofs. To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as: By First Principle Using Chain Rule By First Principle The derivative of ln(x+3) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of ln(x+3) using the first principle, we will consider f(x) = ln(x+3). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = ln(x+3), we write f(x + h) = ln(x + h + 3). Substituting these into equation (1), f'(x) = limₕ→₀ [ln(x + h + 3) - ln(x+3)] / h Using logarithmic properties, we have: f'(x) = limₕ→₀ [ln((x + h + 3)/(x+3))] / h = limₕ→₀ 1/h * ln(1 + h/(x+3)) Using the limit formula, limₕ→₀ ln(1 + u)/u = 1, where u = h/(x+3), f'(x) = 1/(x+3) Hence, proved. Using Chain Rule To prove the differentiation of ln(x+3) using the chain rule, We use the formula: Let u = x+3, so ln(x+3) = ln(u) By chain rule: d/dx [ln(u)] = 1/u * du/dx Substitute u = x+3, du/dx = 1 d/dx (ln(x+3)) = 1/(x+3) * 1 Therefore, d/dx (ln(x+3)) = 1/(x+3).

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Higher-Order Derivatives of ln(x+3)

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky. To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like ln(x+3). For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(x). Similarly, the third derivative, f′′′(x), is the result of the second derivative, and this pattern continues. For the nth derivative of ln(x+3), we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change (continuing for higher-order derivatives).

Special Cases:

When x = -3, the derivative is undefined because ln(x+3) has a vertical asymptote there. When x = 0, the derivative of ln(x+3) = 1/3.

Common Mistakes and How to Avoid Them in Derivatives of ln(x+3)

Students frequently make mistakes when differentiating ln(x+3). These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:

Problem 1

Calculate the derivative of (ln(x+3)·(x²+1))

Okay, lets begin

Here, we have f(x) = ln(x+3)·(x²+1). Using the product rule, f'(x) = u′v + uv′ In the given equation, u = ln(x+3) and v = x²+1. Let’s differentiate each term, u′ = d/dx (ln(x+3)) = 1/(x+3) v′ = d/dx (x²+1) = 2x Substituting into the given equation, f'(x) = (1/(x+3))·(x²+1) + ln(x+3)·(2x) Let’s simplify terms to get the final answer, f'(x) = (x²+1)/(x+3) + 2x ln(x+3) Thus, the derivative of the specified function is (x²+1)/(x+3) + 2x ln(x+3).

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.

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Problem 2

XYZ Corporation is analyzing the growth of their data usage over time. Their usage is modeled by the function y = ln(x+3), where y represents the data usage at time x in months. If x = 9 months, determine the rate of change in data usage.

Okay, lets begin

We have y = ln(x+3) (data usage)...(1) Now, we will differentiate the equation (1) Take the derivative ln(x+3): dy/dx = 1/(x+3) Given x = 9 (substitute this into the derivative) dy/dx = 1/(9+3) = 1/12 Hence, we get the rate of change in data usage at x = 9 months as 1/12.

Explanation

We find the rate of change in data usage at x = 9 months as 1/12, which means that for each additional month, the data usage increases by 1/12 units.

Well explained 👍

Problem 3

Derive the second derivative of the function y = ln(x+3).

Okay, lets begin

The first step is to find the first derivative, dy/dx = 1/(x+3)...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [1/(x+3)] Here we use the chain rule, d²y/dx² = -1/(x+3)² Therefore, the second derivative of the function y = ln(x+3) is -1/(x+3)².

Explanation

We use the step-by-step process, where we start with the first derivative. Using the chain rule, we differentiate 1/(x+3). We then simplify the terms to find the final answer.

Well explained 👍

Problem 4

Prove: d/dx ((ln(x+3))²) = 2 ln(x+3)/(x+3).

Okay, lets begin

Let’s start using the chain rule: Consider y = (ln(x+3))² To differentiate, we use the chain rule: dy/dx = 2 ln(x+3)·d/dx [ln(x+3)] Since the derivative of ln(x+3) is 1/(x+3), dy/dx = 2 ln(x+3)·(1/(x+3)) Substituting y = (ln(x+3))², d/dx ((ln(x+3))²) = 2 ln(x+3)/(x+3) Hence proved.

Explanation

In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace ln(x+3) with its derivative. As a final step, we substitute y = (ln(x+3))² to derive the equation.

Well explained 👍

Problem 5

Solve: d/dx (ln(x+3)/x)

Okay, lets begin

To differentiate the function, we use the quotient rule: d/dx (ln(x+3)/x) = (d/dx (ln(x+3))·x - ln(x+3)·d/dx(x))/x² We will substitute d/dx (ln(x+3)) = 1/(x+3) and d/dx (x) = 1 = (1/(x+3)·x - ln(x+3)·1)/x² = (x/(x+3) - ln(x+3))/x² Therefore, d/dx (ln(x+3)/x) = (x/(x+3) - ln(x+3))/x²

Explanation

In this process, we differentiate the given function using the product rule and quotient rule. As a final step, we simplify the equation to obtain the final result.

Well explained 👍

FAQs on the Derivative of ln(x+3)

1.Find the derivative of ln(x+3).

Using the chain rule for ln(x+3), d/dx (ln(x+3)) = 1/(x+3) (simplified)

2.Can we use the derivative of ln(x+3) in real life?

Yes, we can use the derivative of ln(x+3) in real life in calculating the rate of change of any growth process, especially in fields such as biology, economics, and technology.

3.Is it possible to take the derivative of ln(x+3) at the point where x = -3?

No, x = -3 is a point where ln(x+3) is undefined, so it is impossible to take the derivative at these points (since the function does not exist there).

4.What rule is used to differentiate ln(x+3)/x?

We use the quotient rule to differentiate ln(x+3)/x, d/dx (ln(x+3)/x) = (x.1/(x+3) - ln(x+3).1)/x².

5.Are the derivatives of ln(x+3) and ln(x) the same?

No, they are different. The derivative of ln(x+3) is 1/(x+3), while the derivative of ln(x) is 1/x.

6.Can we find the derivative of the ln(x+3) formula?

To find, consider y = ln(x+3). We use the chain rule: y’ = 1/(x+3).

Important Glossaries for the Derivative of ln(x+3)

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Logarithmic Function: The logarithmic function is a function that uses logarithms, such as ln(x+3). Chain Rule: A rule in calculus for differentiating compositions of functions. Quotient Rule: A method for finding the derivative of a quotient of two functions. Asymptote: A line that a graph approaches but never touches or crosses.

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Jaskaran Singh Saluja

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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.

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