Ohio Assessments for Educators (OAE) Mathematics Practice Exam

Which derivative represents the rate of change of a natural logarithm?

1. 1/x

2. ln(x)/x

3. e^x

4. xln(x)

The correct answer is: 1/x

The derivative of a natural logarithm function is determined by the fundamental rules of differentiation. For the natural logarithm, specifically \( \ln(x) \), the rate of change with respect to \( x \) is given by the formula \( \frac{d}{dx}[\ln(x)] = \frac{1}{x} \). This result indicates how the value of the logarithm changes as \( x \) changes, revealing that as \( x \) increases, the derivative \( \frac{1}{x} \) decreases, showing a diminishing rate of change. When considering the other choices, they do not represent the rate of change of the natural logarithm correctly. For instance, \( \frac{\ln(x)}{x} \) is a more complex function that does not simplify to the derivative of \( \ln(x) \), and its behavior is quite different. The function \( e^x \) serves a different role in calculus, representing the exponential growth rather than a logarithmic rate of change. Finally, \( x \ln(x) \) also represents a product of \( x \) and the logarithm, leading to a more complex expression that again does not match the derivative of \( \ln(x