What is the derivative of y = ln(x)?

The derivative of \( y = \ln(x) \) is \( \frac{1}{x} \).

When we talk about the derivative of a function, we are essentially looking at how the function changes as the input changes. For the natural logarithm function, \( y = \ln(x) \), the derivative tells us the rate at which \( y \) changes with respect to \( x \).

To find the derivative of \( y = \ln(x) \), we use the rules of differentiation. The natural logarithm function is special because its derivative is quite simple. The rule states that if \( y = \ln(x) \), then the derivative \( \frac{dy}{dx} \) is \( \frac{1}{x} \). This means that for any value of \( x \), the rate of change of \( y \) is the reciprocal of \( x \).

For example, if \( x = 2 \), the derivative \( \frac{dy}{dx} \) would be \( \frac{1}{2} \). This tells us that at \( x = 2 \), the function \( y = \ln(x) \) is increasing at a rate of \( \frac{1}{2} \) units per unit increase in \( x \).

Understanding this concept is crucial in calculus, as it helps us analyse the behaviour of logarithmic functions. The derivative \( \frac{1}{x} \) is always positive for \( x > 0 \), indicating that \( \ln(x) \) is an increasing function. This knowledge is useful in various applications, such as solving problems involving growth rates and decay processes.