1. What is the Derivative of Natural Logarithm?

The derivative of the natural logarithm function ln(x) is 1/x. This fundamental calculus result shows how the natural log function changes at any point x > 0.

2. How Does the Calculator Work?

The calculator uses the derivative formula:

Where:

• \( x \) — Input value (must be positive)
• \( \ln x \) — Natural logarithm of x
• \( \frac{1}{x} \) — Reciprocal of x

Explanation: The derivative represents the instantaneous rate of change of the natural log function at any given point x.

3. Importance of Derivative Calculation

Details: Understanding the derivative of ln(x) is crucial in calculus, physics, engineering, and economics for solving optimization problems, growth models, and various differential equations.

4. Using the Calculator

Tips: Enter any positive value for x. The calculator will compute the derivative 1/x. Note that x must be greater than 0 as ln(x) is undefined for x ≤ 0.

5. Frequently Asked Questions (FAQ)

Q1: Why is the derivative of ln(x) equal to 1/x?
A: This result comes from the definition of derivatives and the properties of exponential functions, using the limit definition of the derivative.

Q2: What is the domain of this derivative?
A: The derivative 1/x is defined for all x > 0. The natural log function ln(x) is only defined for positive real numbers.

Q3: Can this calculator handle complex numbers?
A: No, this calculator only works with real positive numbers. The natural log of negative or complex numbers requires complex analysis.

Q4: What are practical applications of this derivative?
A: Applications include calculating growth rates, solving differential equations, optimization problems, and various engineering calculations.

Q5: How is this related to the derivative of other log functions?
A: For logarithms with other bases, the derivative formula is d/dx(log_b x) = 1/(x ln b), where ln b is the natural log of the base.