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Identical expressions
ln(one +x)/(one -x)+2arctg1/x
ln(1 plus x) divide by (1 minus x) plus 2arctg1 divide by x
ln(one plus x) divide by (one minus x) plus 2arctg1 divide by x
ln1+x/1-x+2arctg1/x
ln(1+x) divide by (1-x)+2arctg1 divide by x
Similar expressions
ln(1+x)/(1+x)+2arctg1/x
ln(1+x)/(1-x)-2arctg1/x
ln(1-x)/(1-x)+2arctg1/x

The derivative of the function/ ln(1+x)/(1-x)+2arctg1/x

Derivative of ln(1+x)/(1-x)+2arctg1/x

The solution

You have entered [src]

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

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

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

Detail solution

Differentiate $\frac{\log{\left(x + 1 \right)}}{1 - x} + \frac{2 \operatorname{atan}{\left(1 \right)}}{x}$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \log{\left(x + 1 \right)}$ and $g{\left(x \right)} = 1 - x$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = x + 1$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)$ :
    1. Differentiate $x + 1$ term by term:
      1. The derivative of the constant $1$ is zero.
      2. Apply the power rule: $x$ goes to $1$
      The result is: $1$
    The result of the chain rule is:
    
    $\frac{1}{x + 1}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $1 - x$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. 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: $-1$
    The result is: $-1$
  Now plug in to the quotient rule:
  
  $\frac{\frac{1 - x}{x + 1} + \log{\left(x + 1 \right)}}{\left(1 - x\right)^{2}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
  So, the result is: $- \frac{2 \operatorname{atan}{\left(1 \right)}}{x^{2}}$
The result is: $\frac{\frac{1 - x}{x + 1} + \log{\left(x + 1 \right)}}{\left(1 - x\right)^{2}} - \frac{2 \operatorname{atan}{\left(1 \right)}}{x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

        1           2*log(1 + x)           2           4*atan(1)
----------------- - ------------ + ----------------- + ---------
       2                     3                     2        3   
(1 + x) *(-1 + x)    (-1 + x)      (1 + x)*(-1 + x)        x

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

Simplify

The third derivative [src]

  12*atan(1)           6                   3                    2           6*log(1 + x)
- ---------- - ----------------- - ------------------ - ----------------- + ------------
       4                       3          2         2          3                     4  
      x        (1 + x)*(-1 + x)    (1 + x) *(-1 + x)    (1 + x) *(-1 + x)    (-1 + x)

$$- \frac{2}{\left(x - 1\right) \left(x + 1\right)^{3}} - \frac{3}{\left(x - 1\right)^{2} \left(x + 1\right)^{2}} - \frac{6}{\left(x - 1\right)^{3} \left(x + 1\right)} + \frac{6 \log{\left(x + 1 \right)}}{\left(x - 1\right)^{4}} - \frac{12 \operatorname{atan}{\left(1 \right)}}{x^{4}}$$

Simplify

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Derivative of ln(1+x)/(1-x)+2arctg1/x

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The solution