\( \frac{d y}{d x} \) if \( y=\ln \left(\frac{1}{x}\right) \)

To find \( \frac{d y}{d x} \) when \( y=\ln\left(\frac{1}{x}\right) \), we can first rewrite the expression using the properties of logarithms. We have: \[ y = \ln(1) - \ln(x) = 0 - \ln(x) = -\ln(x). \] Now, differentiating \( y \) with respect to \( x \), we get: \[ \frac{d y}{d x} = -\frac{1}{x}. \] So, the derivative is \( \frac{d y}{d x} = -\frac{1}{x} \). Here’s a little fun fact: the natural logarithm has its roots deep in calculus and is connected to exponential growth processes, appearing everywhere from compound interest to population dynamics! Also, when \( x \) approaches 0 from the right, \( y \) heads towards infinity—how's that for an interesting behavior?