What is the derivative of the natural logarithm function ln(x)?

The derivative of the natural logarithm function, denoted as ln(x), is indeed 1/x. This indicates how the function changes with respect to changes in x. The notation ln(x)' denotes the derivative of ln(x) with respect to x, which emphasizes the transformation of a function into its rate of change.

When you derive ln(x), you determine how steep the curve is at any given point. As x increases, the slope decreases but remains positive, showing that ln(x) is an increasing function. The specific result, 1/x, reveals that the rate of change diminishes as x increases, aligning with the characteristic of logarithmic functions.

The other options either misrepresent the derivative or are not in a standard format for expressing it correctly. Understanding this derivative is foundational for calculus and helps in various applications such as integration, solving growth models, and examining exponential functions.