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The derivative of the function/ log(1/x-1)

Derivative of log(1/x-1)

Function f() - derivative -N order at the point
v

The graph:

from to

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The solution

You have entered [src] 
   /1    \
log|- - 1|
   \x    /
log⁡(−1+1x)\log{\left(-1 + \frac{1}{x} \right)} 
log(1/x - 1)
Detail solution

  1. Let u=−1+1xu = -1 + \frac{1}{x}.

  2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

  3. Then, apply the chain rule. Multiply by ddx(−1+1x)\frac{d}{d x} \left(-1 + \frac{1}{x}\right):

    1. Differentiate −1+1x-1 + \frac{1}{x} term by term:

      1. Apply the power rule: 1x\frac{1}{x} goes to −1x2- \frac{1}{x^{2}}

      2. The derivative of the constant −1-1 is zero.

      The result is: −1x2- \frac{1}{x^{2}}

    The result of the chain rule is:

    −1x2(−1+1x)- \frac{1}{x^{2} \left(-1 + \frac{1}{x}\right)}

  4. Now simplify:

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

The answer is:

1x(x−1)\frac{1}{x \left(x - 1\right)}

The graph
02468-8-6-4-2-1010-2525
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   -1     
----------
 2 /1    \
x *|- - 1|
   \x    /
−1x2(−1+1x)- \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/
−2+1x(1−1x)x3(1−1x)- \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/
2(3+3x(1−1x)+1x2(1−1x)2)x4(1−1x)\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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