Khaleesi Herbert

2021-02-16

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

StrycharzT

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

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