Question

Differentiate the function.

8. \(y = \frac{1}{{\ln x}}\)

Short answer

The differentiation of\(y = \frac{1}{{\ln x}}\) is \(\frac{{ - 1}}{{x{{\left( {\ln x} \right)}^2}}}\).

Step-by-step solution

Formula used

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

Differentiate the given function

Differentiate\(y = \frac{1}{{\ln x}}\)with respect to \(x\) by usingformulas.

\(\begin{aligned}{c}\frac{d}{{dx}}\left( y \right)&= \frac{d}{{dx}}\left( {\frac{1}{{\ln x}}} \right)\\&= \frac{d}{{dx}}{\left( {\ln x} \right)^{ - 1}}\\&= - 1{\left( {\ln x} \right)^{ - 1 - 1}} \cdot \frac{d}{{dx}}\left( {\ln x} \right)\\&= - 1{\left( {\ln x} \right)^{ - 2}} \cdot \frac{1}{x}\\&= \frac{{ - 1}}{{x{{\left( {\ln x} \right)}^2}}}\end{aligned}\)

Hence, the differentiation of \(y = \frac{1}{{\ln x}}\) is \(\frac{{ - 1}}{{x{{\left( {\ln x} \right)}^2}}}\).