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

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

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
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
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. The derivative of is .

      The result is:

    To find :

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. The derivative of is .

      The result is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
      1             log(x) - 1  
-------------- - ---------------
x*(log(x) + 1)                 2
                 x*(log(x) + 1) 
$$- \frac{\log{\left(x \right)} - 1}{x \left(\log{\left(x \right)} + 1\right)^{2}} + \frac{1}{x \left(\log{\left(x \right)} + 1\right)}$$
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))                 
$$\frac{\frac{\left(1 + \frac{2}{\log{\left(x \right)} + 1}\right) \left(\log{\left(x \right)} - 1\right)}{\log{\left(x \right)} + 1} - 1 - \frac{2}{\log{\left(x \right)} + 1}}{x^{2} \left(\log{\left(x \right)} + 1\right)}$$
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))                                    
$$\frac{\frac{3 \left(1 + \frac{2}{\log{\left(x \right)} + 1}\right)}{\log{\left(x \right)} + 1} - \frac{2 \left(\log{\left(x \right)} - 1\right) \left(1 + \frac{3}{\log{\left(x \right)} + 1} + \frac{3}{\left(\log{\left(x \right)} + 1\right)^{2}}\right)}{\log{\left(x \right)} + 1} + 2 + \frac{3}{\log{\left(x \right)} + 1}}{x^{3} \left(\log{\left(x \right)} + 1\right)}$$
The graph
Derivative of (log(x)-1)/(log(x)+1)