Question

Differentiate the function.

7. \(f\left( x \right) = \ln \frac{1}{x}\)

Short answer

The differentiation of\(f\left( x \right) = \ln \frac{1}{x}\) is \( - \frac{1}{x}\).

Step-by-step solution

Formula used

• \(\frac{d}{{dx}}\left( {\ln u} \right) = \frac{1}{u}\frac{{du}}{{dx}}\)
• Chain Rule:\(\frac{d}{{dx}}\left( {{u^n}} \right) = n{u^{n - 1}}\frac{{du}}{{dx}}\)

 Step 2: Differentiate the given function

Differentiate\(f\left( x \right)\)with respect to \(x\) by using formulas.

\(\begin{aligned} {f}'\left( x \right)&=\frac{d}{dx}\left( \ln \frac{1}{x} \right) \\ & =\frac{1}{{1}/{x}\;}\cdot \frac{d}{dx}\left( \frac{1}{x} \right) \\ & =x\cdot \frac{d}{dx}\left( {{x}^{-1}} \right) \\ & =x\cdot \left( -1{{x}^{-1-1}} \right) \\ & =x\left( -{{x}^{-2}} \right) \\ & =\frac{x}{-{{x}^{2}}} \\ & =-\frac{1}{x} \end{aligned}\)

Hence, the differentiation of \(f\left( x \right) = \ln \frac{1}{x}\) is \( - \frac{1}{x}\).