Question

Find the derivatives of the functions. f(x)=\frac{1}{x\ln x}

Derivatives
ANSWERED
asked 2021-05-17
Find the derivatives of the functions. \(\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{{x}{\ln{{x}}}}}}\)

Answers (1)

2021-05-18
The function we want to differentiate is
\(\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{{x}{\ln{{x}}}}}}\)
Apply the quotient rule first to get
\(\displaystyle{f}'{\left({x}\right)}={\frac{{{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({1}\right)}{x}{\ln{{x}}}-{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}{\ln{{x}}}\right)}}}{{{\left({x}{\ln{{x}}}\right)}^{{{2}}}}}}\)
Note that the first term in the numerator is zero because
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({1}\right)}={0}\)
The second term can be computed using the product rule and the derivative of natural logarithm rule:
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\ln{{x}}}\right)}={\frac{{{1}}}{{{x}}}}\)
So we get
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}{\ln{{x}}}\right)}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}\right)}{\ln{{x}}}+{x}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\ln{{x}}}\right)}\)
\(\displaystyle={\ln{{x}}}+{x}{\frac{{{1}}}{{{x}}}}\)
\(\displaystyle={\ln{{x}}}+{1}\)
Substituting back, we get
\(\displaystyle{f}'{\left({x}\right)}={\frac{{-{\left({\ln{{x}}}+{1}\right)}}}{{{\left({x}{\ln{{x}}}\right)}^{{{2}}}}}}\)
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