Derivative of log(1/x-1)

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   /1    \
log|- - 1|
   \x    /

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

log(1/x - 1)

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   -1     
----------
 2 /1    \
x *|- - 1|
   \x    /

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

Simplify

The second derivative [src]

 /        1    \ 
-|2 + ---------| 
 |      /    1\| 
 |    x*|1 - -|| 
 \      \    x// 
-----------------
     3 /    1\   
    x *|1 - -|   
       \    x/

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

Simplify

The third derivative [src]

  /         1            3    \
2*|3 + ----------- + ---------|
  |              2     /    1\|
  |     2 /    1\    x*|1 - -||
  |    x *|1 - -|      \    x/|
  \       \    x/             /
-------------------------------
            4 /    1\          
           x *|1 - -|          
              \    x/

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

Simplify

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Derivative of log(1/x-1)

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The solution