Evaluate the following derivatives. d/(dx)(x^(-ln x))

Question
Derivatives
asked 2021-02-16
Evaluate the following derivatives.
\(\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({x}^{{-{\ln{{x}}}}}\right)}\)

Answers (1)

2021-02-17
Step 1
To evaluate the derivatives: \(\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({x}^{{-{\ln{{x}}}}}\right)}\)
Solution:
Let \(\displaystyle{y}={x}^{{-{\ln{{x}}}}}\)
Taking logarithm on both sides,
\(\displaystyle{\ln{{y}}}={{\ln{{x}}}^{{-{\ln{{x}}}}}}\)
\(\displaystyle{\ln{{y}}}=-{\ln{{x}}}\cdot{\ln{{x}}}\)
\(\displaystyle{\ln{{y}}}=-{\left({\ln{{x}}}\right)}^{{2}}\)
Differentiating with respect to x,
\(\displaystyle\frac{{1}}{{y}}\cdot\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}=-{2}{\ln{{x}}}\cdot\frac{{1}}{{x}}\)
\(\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}=-{2}{y}\frac{{{\ln{{x}}}}}{{x}}\)
\(\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}=\frac{{-{2}{x}^{{-{\ln{{x}}}}}{\ln{{x}}}}}{{x}}\)
Step 2
Hence, required derivative is \(\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}=\frac{{-{2}{x}^{{-{\ln{{x}}}}}{\ln{{x}}}}}{{x}}\).
0

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