Производная atan((1+x)/(1-x))

Вы ввели

    /1 + x\
atan|-----|
    \1 - x/

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

Первая производная

  1      1 + x  
----- + --------
1 - x          2
        (1 - x) 
----------------
             2  
      (1 + x)   
  1 + --------  
             2  
      (1 - x)   

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

Вторая производная

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

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

Третья производная

               /                              2                                                        \
               |         4*(1 + x)   3*(1 + x)                            2                            |
               |     1 - --------- + ----------             2 /    1 + x \                /    1 + x \ |
               |           -1 + x            2     4*(1 + x) *|1 - ------|      4*(1 + x)*|1 - ------| |
  /    1 + x \ |                     (-1 + x)                 \    -1 + x/                \    -1 + x/ |
2*|1 - ------|*|-3 + -------------------------- - -------------------------- - ------------------------|
  \    -1 + x/ |                       2                         2             /            2\         |
               |                (1 + x)           /            2\              |     (1 + x) |         |
               |           1 + ---------          |     (1 + x) |          2   |1 + ---------|*(-1 + x)|
               |                       2          |1 + ---------| *(-1 + x)    |            2|         |
               |               (-1 + x)           |            2|              \    (-1 + x) /         |
               \                                  \    (-1 + x) /                                      /
--------------------------------------------------------------------------------------------------------
                                       /            2\                                                  
                                       |     (1 + x) |         3                                        
                                       |1 + ---------|*(-1 + x)                                         
                                       |            2|                                                  
                                       \    (-1 + x) /                                                  

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

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