What is the derivative of ln(x)?

The derivative of the natural logarithm function, ln(x), is indeed equal to 1/x. This result stems from the fundamental principles of calculus, particularly the rules for differentiating logarithmic functions.

When you differentiate ln(x), you are essentially measuring how the natural logarithm changes as x changes. The function ln(x) is defined only for positive values of x, which makes it a crucial function in many areas of mathematics, especially in calculus and analysis.

To derive this, one can apply the chain rule or use the definition of the derivative, which is the limit of the average rate of change of the function as the interval approaches zero. The behavior of the natural logarithm involves a slow increase as x increases, showing that the rate of change diminishes and is inversely proportional to x. Therefore, for every unit increase in x, the increment in ln(x) becomes progressively smaller, proportional to 1/x.

This characteristic nature of the function means that the derivative, which tells us this rate of change, is accurately represented as 1/x. Hence, the choice stating that the derivative of ln(x) is equal to 1/x is correct and reflects the foundational concepts of calculus concerning logarithmic differentiation.