What is the derivative of ln(x) with respect to x?

• A. 1/x
• B. x
• C. ln(x)
• D. e^x

Answer: (A)

Differentiating a natural logarithm uses the rule d/dx [ln(u)] = u'(x)/u(x). With u(x) = x, the derivative becomes 1/x, since the derivative of x is 1. This is valid for x > 0, because ln(x) is only defined for positive x; outside that domain, the derivative isn’t applicable. So the rate of change of ln(x) with respect to x is 1/x. As x grows, the slope gets smaller, reflecting the logarithm’s increasing but slowly rising nature.