Question

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

Derivatives
Evaluate the following derivatives.
$$\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({x}^{{-{\ln{{x}}}}}\right)}$$

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 $$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}=\frac{{-{2}{x}^{{-{\ln{{x}}}}}{\ln{{x}}}}}{{x}}$$.