Logarithmic Derivative Notes

Published on: September 29, 2026

This Logarithmic Derivative Calculator differentiates natural and common logarithms, and shows the working line by line. The derivative of the natural logarithm is one over x, and a logarithm to another base picks up a division by the logarithm of that base. A constant multiple in front is carried through unchanged. Following the same pattern every time keeps the base factor from being dropped.

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for the natural logarithm.
  3. Apply the rule, keeping the constant multiple in front.

Formula:

\(\frac{d}{dx}\left(\ln\left(x\right)\right) = \frac{1}{x}\)
See Example 1 Hide Example 1

Problem

\(f\left(x\right) = \ln\left(x\right)\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for the natural logarithm.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Identify the logarithmic function and any constant multiple.

In this problem: The function is \(\ln\left(x\right)\).

\(f\left(x\right) = \ln\left(x\right)\)

Step 2

Step 2 - Write the derivative rule for the natural logarithm.

In this problem: The derivative of the natural logarithm is one over x.

\(\frac{d}{dx}\left(\ln\left(x\right)\right) = \frac{1}{x}\)

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front.

In this problem: Replace the logarithm with its derivative and simplify.

\(f'\left(x\right) = \frac{1}{x}\)

Final Answer

\(f'\left(x\right) = \frac{1}{x}\)
See Example 2 Hide Example 2

Problem

\(f\left(x\right) = 3 \ln\left(x\right)\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for the natural logarithm.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Identify the logarithmic function and any constant multiple.

In this problem: The function is \(\ln\left(x\right)\). The constant multiple \(3\) stays in front.

\(f\left(x\right) = 3 \ln\left(x\right)\)

Step 2

Step 2 - Write the derivative rule for the natural logarithm.

In this problem: The derivative of the natural logarithm is one over x.

\(\frac{d}{dx}\left(\ln\left(x\right)\right) = \frac{1}{x}\)

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front.

In this problem: Replace the logarithm with its derivative and simplify.

\(f'\left(x\right) = 3 \cdot \left(\frac{1}{x}\right) = \frac{3}{x}\)

Final Answer

\(f'\left(x\right) = \frac{3}{x}\)
See Example 3 Hide Example 3

Problem

\(f\left(x\right) = 7 \ln\left(x\right)\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for the natural logarithm.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Identify the logarithmic function and any constant multiple.

In this problem: The function is \(\ln\left(x\right)\). The constant multiple \(7\) stays in front.

\(f\left(x\right) = 7 \ln\left(x\right)\)

Step 2

Step 2 - Write the derivative rule for the natural logarithm.

In this problem: The derivative of the natural logarithm is one over x.

\(\frac{d}{dx}\left(\ln\left(x\right)\right) = \frac{1}{x}\)

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front.

In this problem: Replace the logarithm with its derivative and simplify.

\(f'\left(x\right) = 7 \cdot \left(\frac{1}{x}\right) = \frac{7}{x}\)

Final Answer

\(f'\left(x\right) = \frac{7}{x}\)
See Example 4 Hide Example 4

Problem

\(f\left(x\right) = 6 \ln\left(x\right)\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for the natural logarithm.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Identify the logarithmic function and any constant multiple.

In this problem: The function is \(\ln\left(x\right)\). The constant multiple \(6\) stays in front.

\(f\left(x\right) = 6 \ln\left(x\right)\)

Step 2

Step 2 - Write the derivative rule for the natural logarithm.

In this problem: The derivative of the natural logarithm is one over x.

\(\frac{d}{dx}\left(\ln\left(x\right)\right) = \frac{1}{x}\)

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front.

In this problem: Replace the logarithm with its derivative and simplify.

\(f'\left(x\right) = 6 \cdot \left(\frac{1}{x}\right) = \frac{6}{x}\)

Final Answer

\(f'\left(x\right) = \frac{6}{x}\)