Ohio Assessments for Educators (OAE) Mathematics Practice Exam

How is the derivative of ln(x) expressed?

1. 1/x²

2. 1/x

3. ln(x)/x

4. x²

The correct answer is: 1/x

The derivative of the natural logarithm function, ln(x), is derived from the definition of the derivative and the properties of logarithmic functions. The rule states that if you have a function in the form of ln(u), where u is a function of x, the derivative is given by the chain rule as (1/u) * (du/dx). When u is simply x, the expression simplifies to 1/x. Therefore, the derivative of ln(x) is 1/x. This reflects the relationship between the rate of change of the ln function at any point x and the value of x itself. As x increases, the value of the derivative decreases but remains positive for x > 0, indicating that the function ln(x) is increasing but at a decreasing rate. In contrast, the other options represent different mathematical expressions that do not describe the derivative of ln(x). The expression 1/x² signifies a function that decreases more rapidly than 1/x. The expression ln(x)/x combines both logarithmic and linear components but does not accurately represent the derivative. Lastly, x² implies a completely different growth behavior, which is relevant to polynomial functions rather than logarithmic functions.