Logarithmic Derivative Calculator

Published on: July 17, 2026
Final Answer: Free • Full Steps: Plus

This Logarithmic Derivative Calculator helps you differentiate logarithmic functions such as ln(x) or 3ln(x). The derivative of the natural logarithm is one over x, and any constant multiple simply stays in front.

Step-by-step method

  1. Identify the logarithmic function and any constant multiple.
  2. Write the derivative rule for the natural logarithm.
  3. Apply the rule, keeping the constant multiple in front.

Formula:

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

Example 1:

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

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 - 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 - 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:

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

Example 2:

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

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 - 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 - 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:

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