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

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Find the first order derivative with respect to $$ x $$ for $$ y=\ln\left(\frac{1}{x}ight) $$.
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
  $$y^{\prime}=\frac{d}{dx}\left(\ln{\left(\frac{1}{x}ight)}ight)$$
- step1: Use differentiation rules:
  $$y^{\prime}=\frac{d}{dg}\left(\ln{\left(gight)}ight)	imes \frac{d}{dx}\left(\frac{1}{x}ight)$$
- step2: Find the derivative:
  $$y^{\prime}=\frac{1}{g}	imes \frac{d}{dx}\left(\frac{1}{x}ight)$$
- step3: Calculate:
  $$y^{\prime}=\frac{1}{g}	imes \left(-\frac{1}{x^{2}}ight)$$
- step4: Substitute back:
  $$y^{\prime}=\frac{1}{\frac{1}{x}}	imes \left(-\frac{1}{x^{2}}ight)$$
- step5: Calculate:
  $$y^{\prime}=x\left(-\frac{1}{x^{2}}ight)$$
- step6: Multiply the terms:
  $$y^{\prime}=-\frac{1}{x}$$
The derivative of $$ y = \ln\left(\frac{1}{x}ight) $$ with respect to $$ x $$ is $$ \frac{dy}{dx} = -\frac{1}{x} $$.
$$ \frac{dy}{dx} = -\frac{1}{x} $$

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