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Identical expressions
(log(x)- one)/(log(x)+ one)
( logarithm of (x) minus 1) divide by ( logarithm of (x) plus 1)
( logarithm of (x) minus one) divide by ( logarithm of (x) plus one)
logx-1/logx+1
(log(x)-1) divide by (log(x)+1)
Similar expressions
(log(x)+1)/(log(x)+1)
(log(x)-1)/(log(x)-1)

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

Derivative of (log(x)-1)/(log(x)+1)

The solution

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log(x) - 1
----------
log(x) + 1

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

(log(x) - 1)/(log(x) + 1)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \log{\left(x \right)} - 1$ and $g{\left(x \right)} = \log{\left(x \right)} + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $\log{\left(x \right)} - 1$ term by term:
  1. The derivative of the constant $-1$ is zero.
  2. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result is: $\frac{1}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\log{\left(x \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result is: $\frac{1}{x}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

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The first derivative [src]

      1             log(x) - 1  
-------------- - ---------------
x*(log(x) + 1)                 2
                 x*(log(x) + 1)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of (log(x)-1)/(log(x)+1)

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Piecewise:

The solution