Derivative of 1/(x(ln(x)-1))

The solution

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      1       
--------------
x*(log(x) - 1)

$$\frac{1}{x \left(\log{\left(x \right)} - 1\right)}$$

1/(x*(log(x) - 1))

Detail solution

Let $u = x \left(\log{\left(x \right)} - 1\right)$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} x \left(\log{\left(x \right)} - 1\right)$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \log{\left(x \right)} - 1$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $\log{\left(x \right)} - 1$ term by term:
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    2. The derivative of the constant $-1$ is zero.
    The result is: $\frac{1}{x}$
  The result is: $\log{\left(x \right)}$
The result of the chain rule is:

$- \frac{\log{\left(x \right)}}{x^{2} \left(\log{\left(x \right)} - 1\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        1               
- --------------*log(x) 
  x*(log(x) - 1)        
------------------------
     x*(log(x) - 1)

$$- \frac{\frac{1}{x \left(\log{\left(x \right)} - 1\right)} \log{\left(x \right)}}{x \left(\log{\left(x \right)} - 1\right)}$$

Simplify

The second derivative [src]

        log(x)     /         1     \                
-1 + ----------- + |1 + -----------|*log(x) + log(x)
     -1 + log(x)   \    -1 + log(x)/                
----------------------------------------------------
                  3              2                  
                 x *(-1 + log(x))

$$\frac{\left(1 + \frac{1}{\log{\left(x \right)} - 1}\right) \log{\left(x \right)} + \log{\left(x \right)} - 1 + \frac{\log{\left(x \right)}}{\log{\left(x \right)} - 1}}{x^{3} \left(\log{\left(x \right)} - 1\right)^{2}}$$

Simplify

The third derivative [src]

                                                                                                                                   /         1     \       
                                                                                                                                   |1 + -----------|*log(x)
                    4        /         1     \          /          2               3     \            5*log(x)       3*log(x)      \    -1 + log(x)/       
5 - 3*log(x) + ----------- - |1 + -----------|*log(x) - |2 + -------------- + -----------|*log(x) - ----------- - -------------- - ------------------------
               -1 + log(x)   \    -1 + log(x)/          |                 2   -1 + log(x)|          -1 + log(x)                2         -1 + log(x)       
                                                        \    (-1 + log(x))               /                        (-1 + log(x))                            
-----------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                      4              2                                                                     
                                                                     x *(-1 + log(x))

$$\frac{- \left(1 + \frac{1}{\log{\left(x \right)} - 1}\right) \log{\left(x \right)} - \frac{\left(1 + \frac{1}{\log{\left(x \right)} - 1}\right) \log{\left(x \right)}}{\log{\left(x \right)} - 1} - \left(2 + \frac{3}{\log{\left(x \right)} - 1} + \frac{2}{\left(\log{\left(x \right)} - 1\right)^{2}}\right) \log{\left(x \right)} - 3 \log{\left(x \right)} + 5 - \frac{5 \log{\left(x \right)}}{\log{\left(x \right)} - 1} + \frac{4}{\log{\left(x \right)} - 1} - \frac{3 \log{\left(x \right)}}{\left(\log{\left(x \right)} - 1\right)^{2}}}{x^{4} \left(\log{\left(x \right)} - 1\right)^{2}}$$

Simplify

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Derivative of 1/(x(ln(x)-1))

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The solution