Find \( \frac{d y}{d x} \) If: \( y=x \ln \left(\frac{1}{x}\right) \)
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To find \( \frac{d y}{d x} \) for the function \( y = x \ln \left(\frac{1}{x}\right) \), we first simplify the logarithm: \[ \ln\left(\frac{1}{x}\right) = -\ln(x). \] So, we can rewrite \( y \) as: \[ y = x (-\ln(x)) = -x \ln(x). \] Now, we can apply the product rule to differentiate \( y \): \[ \frac{d y}{d x} = -\left( \frac{d}{dx}(x) \cdot \ln(x) + x \cdot \frac{d}{dx}(\ln(x)) \right). \] Calculating the derivatives, we get: \[ \frac{d}{dx}(x) = 1, \] \[ \frac{d}{dx}(\ln(x)) = \frac{1}{x}. \] Substituting these into the product rule gives: \[ \frac{d y}{d x} = -\left( 1 \cdot \ln(x) + x \cdot \frac{1}{x} \right) = -(\ln(x) + 1). \] Thus, the derivative is: \[ \frac{d y}{d x} = -\ln(x) - 1. \]
