Sr Examen

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ln^2(arctg((1+sin(x))/(1-cos(x))))
Derivada de la función/ ln^2(arctg((1+sin(x))/(1-cos(x))))

Derivada de ln^2(arctg((1+sin(x))/(1-cos(x))))

Función f() - derivada -er orden en el punto
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
   2/    /1 + sin(x)\\
log |atan|----------||
    \    \1 - cos(x)//
log⁡(atan⁡(sin⁡(x)+11−cos⁡(x)))2\log{\left(\operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{1 - \cos{\left(x \right)}} \right)} \right)}^{2} 
log(atan((1 + sin(x))/(1 - cos(x))))^2
Gráfica
02468-8-6-4-2-1010-10001000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
  /  cos(x)     (1 + sin(x))*sin(x)\    /    /1 + sin(x)\\
2*|---------- - -------------------|*log|atan|----------||
  |1 - cos(x)                  2   |    \    \1 - cos(x)//
  \                (1 - cos(x))    /                      
----------------------------------------------------------
           /                2\                            
           |    (1 + sin(x)) |     /1 + sin(x)\           
           |1 + -------------|*atan|----------|           
           |                2|     \1 - cos(x)/           
           \    (1 - cos(x)) /
2(cos⁡(x)1−cos⁡(x)−(sin⁡(x)+1)sin⁡(x)(1−cos⁡(x))2)log⁡(atan⁡(sin⁡(x)+11−cos⁡(x)))(1+(sin⁡(x)+1)2(1−cos⁡(x))2)atan⁡(sin⁡(x)+11−cos⁡(x))\frac{2 \left(\frac{\cos{\left(x \right)}}{1 - \cos{\left(x \right)}} - \frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right) \log{\left(\operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{1 - \cos{\left(x \right)}} \right)} \right)}}{\left(1 + \frac{\left(\sin{\left(x \right)} + 1\right)^{2}}{\left(1 - \cos{\left(x \right)}\right)^{2}}\right) \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{1 - \cos{\left(x \right)}} \right)}}
Simplificar
Segunda derivada [src] 
  /                                                                                                                                             2                                            2                                                           2                                     \
  |                                                                                                               /(1 + sin(x))*sin(x)         \               /(1 + sin(x))*sin(x)         \     /     / 1 + sin(x)\\     /(1 + sin(x))*sin(x)         \                  /     / 1 + sin(x)\\|
  |/                                                       2                \                                     |------------------- + cos(x)|               |------------------- + cos(x)| *log|-atan|-----------||   2*|------------------- + cos(x)| *(1 + sin(x))*log|-atan|-----------|||
  ||          (1 + sin(x))*cos(x)   2*cos(x)*sin(x)   2*sin (x)*(1 + sin(x))|    /     / 1 + sin(x)\\             \    -1 + cos(x)             /               \    -1 + cos(x)             /     \     \-1 + cos(x)//     \    -1 + cos(x)             /                  \     \-1 + cos(x)//|
2*||-sin(x) + ------------------- + --------------- + ----------------------|*log|-atan|-----------|| + ---------------------------------------------------- - ------------------------------------------------------- - ----------------------------------------------------------------------|
  ||              -1 + cos(x)         -1 + cos(x)                      2    |    \     \-1 + cos(x)//   /                2 \                                     /                2 \                                                     /                2 \                                 |
  |\                                                      (-1 + cos(x))     /                           |    (1 + sin(x))  |                   / 1 + sin(x)\     |    (1 + sin(x))  |                   / 1 + sin(x)\                     |    (1 + sin(x))  |              2                  |
  |                                                                                                     |1 + --------------|*(-1 + cos(x))*atan|-----------|     |1 + --------------|*(-1 + cos(x))*atan|-----------|                     |1 + --------------|*(-1 + cos(x))                   |
  |                                                                                                     |                 2|                   \-1 + cos(x)/     |                 2|                   \-1 + cos(x)/                     |                 2|                                 |
  \                                                                                                     \    (-1 + cos(x)) /                                     \    (-1 + cos(x)) /                                                     \    (-1 + cos(x)) /                                 /
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                      /                2 \                                                                                                                                                      
                                                                                                                      |    (1 + sin(x))  |                   / 1 + sin(x)\                                                                                                                      
                                                                                                                      |1 + --------------|*(-1 + cos(x))*atan|-----------|                                                                                                                      
                                                                                                                      |                 2|                   \-1 + cos(x)/                                                                                                                      
                                                                                                                      \    (-1 + cos(x)) /
2(((sin⁡(x)+1)cos⁡(x)cos⁡(x)−1+2(sin⁡(x)+1)sin⁡2(x)(cos⁡(x)−1)2−sin⁡(x)+2sin⁡(x)cos⁡(x)cos⁡(x)−1)log⁡(−atan⁡(sin⁡(x)+1cos⁡(x)−1))−2((sin⁡(x)+1)sin⁡(x)cos⁡(x)−1+cos⁡(x))2(sin⁡(x)+1)log⁡(−atan⁡(sin⁡(x)+1cos⁡(x)−1))((sin⁡(x)+1)2(cos⁡(x)−1)2+1)(cos⁡(x)−1)2−((sin⁡(x)+1)sin⁡(x)cos⁡(x)−1+cos⁡(x))2log⁡(−atan⁡(sin⁡(x)+1cos⁡(x)−1))((sin⁡(x)+1)2(cos⁡(x)−1)2+1)(cos⁡(x)−1)atan⁡(sin⁡(x)+1cos⁡(x)−1)+((sin⁡(x)+1)sin⁡(x)cos⁡(x)−1+cos⁡(x))2((sin⁡(x)+1)2(cos⁡(x)−1)2+1)(cos⁡(x)−1)atan⁡(sin⁡(x)+1cos⁡(x)−1))((sin⁡(x)+1)2(cos⁡(x)−1)2+1)(cos⁡(x)−1)atan⁡(sin⁡(x)+1cos⁡(x)−1)\frac{2 \left(\left(\frac{\left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} + \frac{2 \left(\sin{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}} - \sin{\left(x \right)} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \log{\left(- \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)} \right)} - \frac{2 \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} + \cos{\left(x \right)}\right)^{2} \left(\sin{\left(x \right)} + 1\right) \log{\left(- \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)} \right)}}{\left(\frac{\left(\sin{\left(x \right)} + 1\right)^{2}}{\left(\cos{\left(x \right)} - 1\right)^{2}} + 1\right) \left(\cos{\left(x \right)} - 1\right)^{2}} - \frac{\left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} + \cos{\left(x \right)}\right)^{2} \log{\left(- \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)} \right)}}{\left(\frac{\left(\sin{\left(x \right)} + 1\right)^{2}}{\left(\cos{\left(x \right)} - 1\right)^{2}} + 1\right) \left(\cos{\left(x \right)} - 1\right) \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)}} + \frac{\left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} + \cos{\left(x \right)}\right)^{2}}{\left(\frac{\left(\sin{\left(x \right)} + 1\right)^{2}}{\left(\cos{\left(x \right)} - 1\right)^{2}} + 1\right) \left(\cos{\left(x \right)} - 1\right) \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)}}\right)}{\left(\frac{\left(\sin{\left(x \right)} + 1\right)^{2}}{\left(\cos{\left(x \right)} - 1\right)^{2}} + 1\right) \left(\cos{\left(x \right)} - 1\right) \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)}}
Simplificar
Tercera derivada [src] 
  /                                                                                                                                                                                                                                                                                                                     /                                            2                        2    2                                  \                                                                                                                        /                                                       2                \                                                                                                                           /                                                       2                \                                                            /                                                       2                \                                                                                                 \
  |                                                                                                                                                                                                            3                                                  3                      /(1 + sin(x))*sin(x)         \ |   2                            (1 + sin(x)) *cos(x)   3*(1 + sin(x)) *sin (x)   4*(1 + sin(x))*cos(x)*sin(x)|    /     / 1 + sin(x)\\                                   3                             /(1 + sin(x))*sin(x)         \ |          (1 + sin(x))*cos(x)   2*cos(x)*sin(x)   2*sin (x)*(1 + sin(x))|                                                 3                                          /(1 + sin(x))*sin(x)         \ |          (1 + sin(x))*cos(x)   2*cos(x)*sin(x)   2*sin (x)*(1 + sin(x))|    /     / 1 + sin(x)\\     /(1 + sin(x))*sin(x)         \ |          (1 + sin(x))*cos(x)   2*cos(x)*sin(x)   2*sin (x)*(1 + sin(x))|    /     / 1 + sin(x)\\                                   3                                     |
  |                                                                                                                                                                              /(1 + sin(x))*sin(x)         \                     /(1 + sin(x))*sin(x)         \                     2*|------------------- + cos(x)|*|cos (x) - (1 + sin(x))*sin(x) + -------------------- + ----------------------- + ----------------------------|*log|-atan|-----------||     /(1 + sin(x))*sin(x)         \     /     / 1 + sin(x)\\   3*|------------------- + cos(x)|*|-sin(x) + ------------------- + --------------- + ----------------------|                 2 /(1 + sin(x))*sin(x)         \     /     / 1 + sin(x)\\   4*(1 + sin(x))*|------------------- + cos(x)|*|-sin(x) + ------------------- + --------------- + ----------------------|*log|-atan|-----------||   3*|------------------- + cos(x)|*|-sin(x) + ------------------- + --------------- + ----------------------|*log|-atan|-----------||     /(1 + sin(x))*sin(x)         \                  /     / 1 + sin(x)\\|
  |/                2             2                                  3                        2                                         \                                      3*|------------------- + cos(x)|                   6*|------------------- + cos(x)| *(1 + sin(x))         \    -1 + cos(x)             / |                                    -1 + cos(x)                          2               -1 + cos(x)         |    \     \-1 + cos(x)//   2*|------------------- + cos(x)| *log|-atan|-----------||     \    -1 + cos(x)             / |              -1 + cos(x)         -1 + cos(x)                      2    |   8*(1 + sin(x)) *|------------------- + cos(x)| *log|-atan|-----------||                  \    -1 + cos(x)             / |              -1 + cos(x)         -1 + cos(x)                      2    |    \     \-1 + cos(x)//     \    -1 + cos(x)             / |              -1 + cos(x)         -1 + cos(x)                      2    |    \     \-1 + cos(x)//   6*|------------------- + cos(x)| *(1 + sin(x))*log|-atan|-----------|||
  ||           3*sin (x)     3*cos (x)    (1 + sin(x))*sin(x)   6*sin (x)*(1 + sin(x))   6*sin (x)*cos(x)   6*(1 + sin(x))*cos(x)*sin(x)|    /     / 1 + sin(x)\\                \    -1 + cos(x)             /                     \    -1 + cos(x)             /                                                      \                                                            (-1 + cos(x))                                    /                             \    -1 + cos(x)             /     \     \-1 + cos(x)//                                    \                                                      (-1 + cos(x))     /                   \    -1 + cos(x)             /     \     \-1 + cos(x)//                                                 \                                                      (-1 + cos(x))     /                                                            \                                                      (-1 + cos(x))     /                             \    -1 + cos(x)             /                  \     \-1 + cos(x)//|
2*||-cos(x) - ----------- + ----------- - ------------------- + ---------------------- + ---------------- + ----------------------------|*log|-atan|-----------|| - ------------------------------------------------------- - ------------------------------------------------------ - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------ - ----------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------|
  ||          -1 + cos(x)   -1 + cos(x)       -1 + cos(x)                        3                     2                        2       |    \     \-1 + cos(x)//                       2                                                         2                                                                                                      /                2 \                                                                                                          2                                                                  /                2 \                                                                                                    2                                                                                         /                2 \                                                                                                                 /                2 \                                                                                                      2                                         |
  |\                                                                (-1 + cos(x))         (-1 + cos(x))            (-1 + cos(x))        /                           /                2 \                                      /                2 \                                                                                                       |    (1 + sin(x))  |              2                                                                       /                2 \                                                                   |    (1 + sin(x))  |                   / 1 + sin(x)\                                                /                2 \                                                                                          |    (1 + sin(x))  |              2                                                                                                  |    (1 + sin(x))  |                   / 1 + sin(x)\                                                  /                2 \                                          |
  |                                                                                                                                                                 |    (1 + sin(x))  |               2     2/ 1 + sin(x)\   |    (1 + sin(x))  |               3     / 1 + sin(x)\                                                                     |1 + --------------|*(-1 + cos(x))                                                                        |    (1 + sin(x))  |               2     2/ 1 + sin(x)\                                |1 + --------------|*(-1 + cos(x))*atan|-----------|                                                |    (1 + sin(x))  |               4                                                                          |1 + --------------|*(-1 + cos(x))                                                                                                   |1 + --------------|*(-1 + cos(x))*atan|-----------|                                                  |    (1 + sin(x))  |               3     / 1 + sin(x)\        |
  |                                                                                                                                                                 |1 + --------------| *(-1 + cos(x)) *atan |-----------|   |1 + --------------| *(-1 + cos(x)) *atan|-----------|                                                                     |                 2|                                                                                      |1 + --------------| *(-1 + cos(x)) *atan |-----------|                                |                 2|                   \-1 + cos(x)/                                                |1 + --------------| *(-1 + cos(x))                                                                           |                 2|                                                                                                                 |                 2|                   \-1 + cos(x)/                                                  |1 + --------------| *(-1 + cos(x)) *atan|-----------|        |
  |                                                                                                                                                                 |                 2|                      \-1 + cos(x)/   |                 2|                     \-1 + cos(x)/                                                                     \    (-1 + cos(x)) /                                                                                      |                 2|                      \-1 + cos(x)/                                \    (-1 + cos(x)) /                                                                                |                 2|                                                                                          \    (-1 + cos(x)) /                                                                                                                 \    (-1 + cos(x)) /                                                                                  |                 2|                     \-1 + cos(x)/        |
  \                                                                                                                                                                 \    (-1 + cos(x)) /                                      \    (-1 + cos(x)) /                                                                                                                                                                                                                 \    (-1 + cos(x)) /                                                                                                                                                                       \    (-1 + cos(x)) /                                                                                                                                                                                                                                                                                                                                     \    (-1 + cos(x)) /                                          /
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 /                2 \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |    (1 + sin(x))  |                   / 1 + sin(x)\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |1 + --------------|*(-1 + cos(x))*atan|-----------|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |                 2|                   \-1 + cos(x)/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 \    (-1 + cos(x)) /
2((−(sin⁡(x)+1)sin⁡(x)cos⁡(x)−1+6(sin⁡(x)+1)sin⁡(x)cos⁡(x)(cos⁡(x)−1)2+6(sin⁡(x)+1)sin⁡3(x)(cos⁡(x)−1)3−cos⁡(x)−3sin⁡2(x)cos⁡(x)−1+3cos⁡2(x)cos⁡(x)−1+6sin⁡2(x)cos⁡(x)(cos⁡(x)−1)2)log⁡(−atan⁡(sin⁡(x)+1cos⁡(x)−1))−4((sin⁡(x)+1)sin⁡(x)cos⁡(x)−1+cos⁡(x))(sin⁡(x)+1)((sin⁡(x)+1)cos⁡(x)cos⁡(x)−1+2(sin⁡(x)+1)sin⁡2(x)(cos⁡(x)−1)2−sin⁡(x)+2sin⁡(x)cos⁡(x)cos⁡(x)−1)log⁡(−atan⁡(sin⁡(x)+1cos⁡(x)−1))((sin⁡(x)+1)2(cos⁡(x)−1)2+1)(cos⁡(x)−1)2−3((sin⁡(x)+1)sin⁡(x)cos⁡(x)−1+cos⁡(x))((sin⁡(x)+1)cos⁡(x)cos⁡(x)−1+2(sin⁡(x)+1)sin⁡2(x)(cos⁡(x)−1)2−sin⁡(x)+2sin⁡(x)cos⁡(x)cos⁡(x)−1)log⁡(−atan⁡(sin⁡(x)+1cos⁡(x)−1))((sin⁡(x)+1)2(cos⁡(x)−1)2+1)(cos⁡(x)−1)atan⁡(sin⁡(x)+1cos⁡(x)−1)+3((sin⁡(x)+1)sin⁡(x)cos⁡(x)−1+cos⁡(x))((sin⁡(x)+1)cos⁡(x)cos⁡(x)−1+2(sin⁡(x)+1)sin⁡2(x)(cos⁡(x)−1)2−sin⁡(x)+2sin⁡(x)cos⁡(x)cos⁡(x)−1)((sin⁡(x)+1)2(cos⁡(x)−1)2+1)(cos⁡(x)−1)atan⁡(sin⁡(x)+1cos⁡(x)−1)−2((sin⁡(x)+1)sin⁡(x)cos⁡(x)−1+cos⁡(x))((sin⁡(x)+1)2cos⁡(x)cos⁡(x)−1+3(sin⁡(x)+1)2sin⁡2(x)(cos⁡(x)−1)2−(sin⁡(x)+1)sin⁡(x)+4(sin⁡(x)+1)sin⁡(x)cos⁡(x)cos⁡(x)−1+cos⁡2(x))log⁡(−atan⁡(sin⁡(x)+1cos⁡(x)−1))((sin⁡(x)+1)2(cos⁡(x)−1)2+1)(cos⁡(x)−1)2+8((sin⁡(x)+1)sin⁡(x)cos⁡(x)−1+cos⁡(x))3(sin⁡(x)+1)2log⁡(−atan⁡(sin⁡(x)+1cos⁡(x)−1))((sin⁡(x)+1)2(cos⁡(x)−1)2+1)2(cos⁡(x)−1)4+6((sin⁡(x)+1)sin⁡(x)cos⁡(x)−1+cos⁡(x))3(sin⁡(x)+1)log⁡(−atan⁡(sin⁡(x)+1cos⁡(x)−1))((sin⁡(x)+1)2(cos⁡(x)−1)2+1)2(cos⁡(x)−1)3atan⁡(sin⁡(x)+1cos⁡(x)−1)−6((sin⁡(x)+1)sin⁡(x)cos⁡(x)−1+cos⁡(x))3(sin⁡(x)+1)((sin⁡(x)+1)2(cos⁡(x)−1)2+1)2(cos⁡(x)−1)3atan⁡(sin⁡(x)+1cos⁡(x)−1)+2((sin⁡(x)+1)sin⁡(x)cos⁡(x)−1+cos⁡(x))3log⁡(−atan⁡(sin⁡(x)+1cos⁡(x)−1))((sin⁡(x)+1)2(cos⁡(x)−1)2+1)2(cos⁡(x)−1)2atan⁡2(sin⁡(x)+1cos⁡(x)−1)−3((sin⁡(x)+1)sin⁡(x)cos⁡(x)−1+cos⁡(x))3((sin⁡(x)+1)2(cos⁡(x)−1)2+1)2(cos⁡(x)−1)2atan⁡2(sin⁡(x)+1cos⁡(x)−1))((sin⁡(x)+1)2(cos⁡(x)−1)2+1)(cos⁡(x)−1)atan⁡(sin⁡(x)+1cos⁡(x)−1)\frac{2 \left(\left(- \frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} + \frac{6 \left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}} + \frac{6 \left(\sin{\left(x \right)} + 1\right) \sin^{3}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{3}} - \cos{\left(x \right)} - \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1} + \frac{3 \cos^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1} + \frac{6 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}}\right) \log{\left(- \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)} \right)} - \frac{4 \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} + \cos{\left(x \right)}\right) \left(\sin{\left(x \right)} + 1\right) \left(\frac{\left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} + \frac{2 \left(\sin{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}} - \sin{\left(x \right)} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \log{\left(- \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)} \right)}}{\left(\frac{\left(\sin{\left(x \right)} + 1\right)^{2}}{\left(\cos{\left(x \right)} - 1\right)^{2}} + 1\right) \left(\cos{\left(x \right)} - 1\right)^{2}} - \frac{3 \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} + \cos{\left(x \right)}\right) \left(\frac{\left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} + \frac{2 \left(\sin{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}} - \sin{\left(x \right)} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \log{\left(- \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)} \right)}}{\left(\frac{\left(\sin{\left(x \right)} + 1\right)^{2}}{\left(\cos{\left(x \right)} - 1\right)^{2}} + 1\right) \left(\cos{\left(x \right)} - 1\right) \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)}} + \frac{3 \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} + \cos{\left(x \right)}\right) \left(\frac{\left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} + \frac{2 \left(\sin{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}} - \sin{\left(x \right)} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} - 1}\right)}{\left(\frac{\left(\sin{\left(x \right)} + 1\right)^{2}}{\left(\cos{\left(x \right)} - 1\right)^{2}} + 1\right) \left(\cos{\left(x \right)} - 1\right) \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)}} - \frac{2 \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} + \cos{\left(x \right)}\right) \left(\frac{\left(\sin{\left(x \right)} + 1\right)^{2} \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} + \frac{3 \left(\sin{\left(x \right)} + 1\right)^{2} \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}} - \left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)} + \frac{4 \left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} + \cos^{2}{\left(x \right)}\right) \log{\left(- \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)} \right)}}{\left(\frac{\left(\sin{\left(x \right)} + 1\right)^{2}}{\left(\cos{\left(x \right)} - 1\right)^{2}} + 1\right) \left(\cos{\left(x \right)} - 1\right)^{2}} + \frac{8 \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} + \cos{\left(x \right)}\right)^{3} \left(\sin{\left(x \right)} + 1\right)^{2} \log{\left(- \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)} \right)}}{\left(\frac{\left(\sin{\left(x \right)} + 1\right)^{2}}{\left(\cos{\left(x \right)} - 1\right)^{2}} + 1\right)^{2} \left(\cos{\left(x \right)} - 1\right)^{4}} + \frac{6 \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} + \cos{\left(x \right)}\right)^{3} \left(\sin{\left(x \right)} + 1\right) \log{\left(- \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)} \right)}}{\left(\frac{\left(\sin{\left(x \right)} + 1\right)^{2}}{\left(\cos{\left(x \right)} - 1\right)^{2}} + 1\right)^{2} \left(\cos{\left(x \right)} - 1\right)^{3} \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)}} - \frac{6 \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} + \cos{\left(x \right)}\right)^{3} \left(\sin{\left(x \right)} + 1\right)}{\left(\frac{\left(\sin{\left(x \right)} + 1\right)^{2}}{\left(\cos{\left(x \right)} - 1\right)^{2}} + 1\right)^{2} \left(\cos{\left(x \right)} - 1\right)^{3} \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)}} + \frac{2 \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} + \cos{\left(x \right)}\right)^{3} \log{\left(- \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)} \right)}}{\left(\frac{\left(\sin{\left(x \right)} + 1\right)^{2}}{\left(\cos{\left(x \right)} - 1\right)^{2}} + 1\right)^{2} \left(\cos{\left(x \right)} - 1\right)^{2} \operatorname{atan}^{2}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)}} - \frac{3 \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} + \cos{\left(x \right)}\right)^{3}}{\left(\frac{\left(\sin{\left(x \right)} + 1\right)^{2}}{\left(\cos{\left(x \right)} - 1\right)^{2}} + 1\right)^{2} \left(\cos{\left(x \right)} - 1\right)^{2} \operatorname{atan}^{2}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)}}\right)}{\left(\frac{\left(\sin{\left(x \right)} + 1\right)^{2}}{\left(\cos{\left(x \right)} - 1\right)^{2}} + 1\right) \left(\cos{\left(x \right)} - 1\right) \operatorname{atan}{\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} - 1} \right)}}
Simplificar
Gráfico
Derivada de ln^2(arctg((1+sin(x))/(1-cos(x))))
    © Sr Examen – Калькулятор