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Step-by-step Solution

Find the derivative of 1/(xln(x))

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Answer

$\frac{-\left(\ln\left(x\right)+x\frac{1}{x}\right)}{x^2\ln\left(x\right)^2}$

Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(\frac{1}{x\cdot \ln\left(x\right)}\right)$
1

Applying the quotient rule which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{x\frac{d}{dx}\left(1\right)\ln\left(x\right)-\frac{d}{dx}\left(x\ln\left(x\right)\right)}{x^2\ln\left(x\right)^2}$
2

The derivative of the constant function is equal to zero

$\frac{0x\ln\left(x\right)-\frac{d}{dx}\left(x\ln\left(x\right)\right)}{x^2\ln\left(x\right)^2}$

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Answer

$\frac{-\left(\ln\left(x\right)+x\frac{1}{x}\right)}{x^2\ln\left(x\right)^2}$
$\frac{d}{dx}\left(\frac{1}{x\cdot \ln\left(x\right)}\right)$

Main topic:

Differential calculus

Used formulas:

5. See formulas

Time to solve it:

~ 0.8 seconds