Find \( \frac{d y}{d x} \) If: \( y=x \ln \left(\frac{1}{x}\right) \)

To find \( \frac{d y}{d x} \) for the function \( y = x \ln\left(\frac{1}{x}\right) \), we start by rewriting the expression inside the logarithm: \[ y = x \ln\left(\frac{1}{x}\right) = x \ln(1) - x \ln(x) = -x \ln(x) \] Now, we will differentiate \( y = -x \ln(x) \) using the product rule. Recall that the product rule states: \[ \frac{d}{dx}(u \cdot v) = u'v + uv' \] In this case, let \( u = -x \) and \( v = \ln(x) \). We have: \[ u' = -1 \quad \text{and} \quad v' = \frac{1}{x} \] Now we apply the product rule: \[ \frac{d y}{d x} = u'v + uv' = (-1) \ln(x) + (-x) \cdot \frac{1}{x} \] This simplifies to: \[ \frac{d y}{d x} = -\ln(x) - 1 \] Therefore, the derivative is: \[ \frac{d y}{d x} = -\ln(x) - 1 \]