Derivative of ln(x^3)
2026-02-28 23:51 Diff
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Last updated on August 5, 2025

We use the derivative of ln(x^3), which is (3/x), 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 3/x. 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 a composition of logarithmic and power functions. Chain Rule: Rule for differentiating composite functions like ln(x^3). Natural Logarithm: ln(x) is the logarithm to the base e.

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)) = 3/x (or) (ln(x^3))' = 3/x The formula applies to all x > 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 logarithmic identities along with the rules of differentiation. There are several methods we use to prove this, such as: By First Principle Using Chain Rule We will now demonstrate that the differentiation of ln(x^3) results in 3/x using the above-mentioned methods: Using Chain Rule To prove the differentiation of ln(x^3) using the chain rule, Consider f(x) = x^3 and g(x) = ln(x) So, ln(f(x)) = ln(x^3) = 3ln(x) By chain rule: d/dx [ln(f(x))] = f'(x)/f(x) Let’s substitute f(x) = x^3 in the formula, d/dx (ln(x^3)) = d/dx (3ln(x)) = 3 * d/dx (ln(x)) = 3 * (1/x) = 3/x Therefore, the derivative of ln(x^3) is 3/x.

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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 approaches 0, the derivative is undefined because ln(x) is undefined for x ≤ 0. When x = 1, the derivative of ln(x^3) = 3/1, which is 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) + x^2.

Okay, lets begin

Here, we have f(x) = ln(x^3) + x^2. Differentiating each term separately, d/dx (ln(x^3)) = 3/x d/dx (x^2) = 2x So, f'(x) = 3/x + 2x Thus, the derivative of the specified function is 3/x + 2x.

Explanation

We find the derivative of the given function by differentiating each term separately and then combining the results to get the final answer.

Well explained 👍

Problem 2

A cylindrical tank is being filled with water, and its volume V is given by V = ln(x^3), where x is the height of water in the tank. If x = 3 meters, find the rate of change of volume with respect to height.

Okay, lets begin

We have V = ln(x^3)...(1) Now, we will differentiate the equation (1) with respect to x, dV/dx = 3/x Given x = 3, substitute this into the derivative, dV/dx = 3/3 = 1 Hence, the rate of change of volume with respect to height at x = 3 meters is 1.

Explanation

We find the rate of change of the volume at x = 3 meters, which means that at this point, the volume increases linearly with the height.

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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 = 3/x...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [3/x] = -3/x² Therefore, the second derivative of the function y = ln(x^3) is -3/x².

Explanation

We use the step-by-step process, where we start with the first derivative and then differentiate again to find the second derivative, ensuring correct application of differentiation rules.

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Problem 4

Prove: d/dx (ln(x^3)) = 3/x using logarithmic properties.

Okay, lets begin

Let’s start using logarithmic properties: Consider y = ln(x^3) Using the property of logarithms, y = 3ln(x) Differentiating, dy/dx = 3 * d/dx (ln(x)) = 3 * (1/x) = 3/x Hence proved.

Explanation

In this step-by-step process, we use the property of logarithms to simplify ln(x^3) and then differentiate using basic differentiation rules.

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)) = 3/x and d/dx(x) = 1 = (3/x * x - ln(x^3)) / x² = (3 - ln(x^3)) / x² Therefore, d/dx (ln(x^3)/x) = (3 - ln(x^3)) / x²

Explanation

In this process, we differentiate the given function using the quotient rule and 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 on ln(x^3) gives: d/dx (ln(x^3)) = 3/x (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 to calculate the rate of change of growth scenarios, especially in fields such as biology, economics, and environmental science.

3.Is it possible to take the derivative of ln(x^3) at x = 0?

No, x = 0 is a point where ln(x) is undefined, so it is impossible to take the derivative at this point (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 * d/dx (ln(x^3)) - ln(x^3) * 1) / x².

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

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

6.Can we find the derivative of ln(x^3) using a different method?

Yes, we can use the property of logarithms to simplify ln(x^3) to 3ln(x) and then differentiate to obtain the same result, 3/x.

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: A function involving the logarithm, such as ln(x). Chain Rule: A rule used to differentiate composite functions. Natural Logarithm: The logarithm to the base e, denoted as ln(x). Quotient Rule: A rule for differentiating the quotient of two functions.

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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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