Derivative of (lnx+x)/(lnx-x)

The solution

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

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

d /log(x) + x\
--|----------|
dx\log(x) - x/

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (lnx+x)/(lnx-x)

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