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Derivada de:
Derivada de 1-4*sin(x)
Derivada de ч^(-10)
Derivada de ч+6
Derivada de ч+1
Expresiones idénticas
((x-tgx^ uno / dos)/(x+tgx^ uno / dos))^ dos
((x menos tgx en el grado 1 dividir por 2) dividir por (x más tgx en el grado 1 dividir por 2)) al cuadrado
((x menos tgx en el grado uno dividir por dos) dividir por (x más tgx en el grado uno dividir por dos)) en el grado dos
((x-tgx1/2)/(x+tgx1/2))2
x-tgx1/2/x+tgx1/22
((x-tgx^1/2)/(x+tgx^1/2))²
((x-tgx en el grado 1/2)/(x+tgx en el grado 1/2)) en el grado 2
x-tgx^1/2/x+tgx^1/2^2
((x-tgx^1 dividir por 2) dividir por (x+tgx^1 dividir por 2))^2
Expresiones semejantes
((x-tgx^1/2)/(x-tgx^1/2))^2
((x+tgx^1/2)/(x+tgx^1/2))^2

Derivada de la función/ ((x-tgx^1/2)/(x+tgx^1/2))^2

Derivada de ((x-tgx^1/2)/(x+tgx^1/2))^2

Solución

Ha introducido [src]

                2
/      ________\ 
|x - \/ tan(x) | 
|--------------| 
|      ________| 
\x + \/ tan(x) /

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

((x - sqrt(tan(x)))/(x + sqrt(tan(x))))^2

Solución detallada

Sustituimos $u = \frac{x - \sqrt{\tan{\left(x \right)}}}{x + \sqrt{\tan{\left(x \right)}}}$ .
Según el principio, aplicamos: $u^{2}$ tenemos $2 u$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{x - \sqrt{\tan{\left(x \right)}}}{x + \sqrt{\tan{\left(x \right)}}}$ :
1. Se aplica la regla de la derivada parcial:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = x - \sqrt{\tan{\left(x \right)}}$ y $g{\left(x \right)} = x + \sqrt{\tan{\left(x \right)}}$ .
  
  Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
  1. diferenciamos $x - \sqrt{\tan{\left(x \right)}}$ miembro por miembro:
    1. Según el principio, aplicamos: $x$ tenemos $1$
    2. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
      1. Sustituimos $u = \tan{\left(x \right)}$ .
      2. Según el principio, aplicamos: $\sqrt{u}$ tenemos $\frac{1}{2 \sqrt{u}}$
      3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \tan{\left(x \right)}$ :
        
        Reescribimos las funciones para diferenciar:
        
        $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
        
        Se aplica la regla de la derivada parcial:
        
        $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
        
        $f{\left(x \right)} = \sin{\left(x \right)}$ y $g{\left(x \right)} = \cos{\left(x \right)}$ .
        
        Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
        
        La derivada del seno es igual al coseno:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
        
        Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
        
        La derivada del coseno es igual a menos el seno:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
        
        Ahora aplicamos la regla de la derivada de una divesión:
        
        $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
        
        Como resultado de la secuencia de reglas:
        
        $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} \sqrt{\tan{\left(x \right)}}}$
      Entonces, como resultado: $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} \sqrt{\tan{\left(x \right)}}}$
    Como resultado de: $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} \sqrt{\tan{\left(x \right)}}} + 1$
  Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
  1. diferenciamos $x + \sqrt{\tan{\left(x \right)}}$ miembro por miembro:
    1. Según el principio, aplicamos: $x$ tenemos $1$
    2. Sustituimos $u = \tan{\left(x \right)}$ .
    3. Según el principio, aplicamos: $\sqrt{u}$ tenemos $\frac{1}{2 \sqrt{u}}$
    4. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \tan{\left(x \right)}$ :
      1. $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
      Como resultado de la secuencia de reglas:
      
      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} \sqrt{\tan{\left(x \right)}}}$
    Como resultado de: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} \sqrt{\tan{\left(x \right)}}} + 1$
  Ahora aplicamos la regla de la derivada de una divesión:
  
  $\frac{- \left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} \sqrt{\tan{\left(x \right)}}} + 1\right) + \left(x + \sqrt{\tan{\left(x \right)}}\right) \left(- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} \sqrt{\tan{\left(x \right)}}} + 1\right)}{\left(x + \sqrt{\tan{\left(x \right)}}\right)^{2}}$
Como resultado de la secuencia de reglas:

$\frac{2 \left(x - \sqrt{\tan{\left(x \right)}}\right) \left(- \left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} \sqrt{\tan{\left(x \right)}}} + 1\right) + \left(x + \sqrt{\tan{\left(x \right)}}\right) \left(- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} \sqrt{\tan{\left(x \right)}}} + 1\right)\right)}{\left(x + \sqrt{\tan{\left(x \right)}}\right)^{3}}$
Simplificamos:

$- \frac{2 \left(x - \sin{\left(2 x \right)}\right) \left(x - \sqrt{\tan{\left(x \right)}}\right)}{\left(x + \sqrt{\tan{\left(x \right)}}\right)^{3} \cos^{2}{\left(x \right)} \sqrt{\tan{\left(x \right)}}}$

Respuesta:

$- \frac{2 \left(x - \sin{\left(2 x \right)}\right) \left(x - \sqrt{\tan{\left(x \right)}}\right)}{\left(x + \sqrt{\tan{\left(x \right)}}\right)^{3} \cos^{2}{\left(x \right)} \sqrt{\tan{\left(x \right)}}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        /                                                                                                         2                                                                                                                                             \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                /                   /           2   \                 \ |                                                                                        /           2   \                  2                   2               /                        2   \     /            2   \ /           2   \                 |                                                            2                                                                              2                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |                   |    1 + tan (x)| /      ________\| |                                                                                        |    1 + tan (x)|  /      ________\    /      ________\  /       2   \ |      ________   1 + tan (x)|     |     1 + tan (x)| |    1 + tan (x)| /      ________\|     /                   /           2   \                 \                        /                   /           2   \                 \                                                             /                   /           2   \                 \                                          /                   /           2   \                 \                                                                 /                   /           2   \                 \
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |                   |2 + -----------|*\x - \/ tan(x) /| |                  2                                                                   3*|2 + -----------| *\x - \/ tan(x) /    \x - \/ tan(x) / *\1 + tan (x)/*|- 4*\/ tan(x)  + -----------|   4*|-2 + -----------|*|2 + -----------|*\x - \/ tan(x) /|     |                   |    1 + tan (x)| /      ________\|                        |                   |    1 + tan (x)| /      ________\|                                                             |                   |    1 + tan (x)| /      ________\|                                          |                   |    1 + tan (x)| /      ________\|                                                                 |                   |    1 + tan (x)| /      ________\|
                   /                                                                                       2                                         3                                                   /                                                2\                                                                                                                                                                                                                           \                                                             /                                                                  2                                                                                                                          \                                                                                                            |            2      |       ________|                 | |/            2   \                                   /                        2   \     |       ________|                                                      |                     3/2    |     |        ________| |       ________|                 |     |                   |2 + -----------|*\x - \/ tan(x) /|                        |                   |2 + -----------|*\x - \/ tan(x) /|                                          2                  |                   |2 + -----------|*\x - \/ tan(x) /|                                          |                   |2 + -----------|*\x - \/ tan(x) /|                                                                 |                   |2 + -----------|*\x - \/ tan(x) /|
                   |                                                                      /           2   \  /            2   \     /           2   \                                                    |                 /       2   \     /       2   \ |                   /            2   \ /                        2   \                   /           2   \ /                        2   \                   /           2   \                  /                        2   \|     /                   /           2   \                 \ |                                                 /           2   \                       /            2   \ /           2   \                                  /                        2   \|                                                /                   /           2   \                 \     |     1 + tan (x)   \     \/ tan(x) /                 | ||     1 + tan (x)|    /       2   \ /      ________\ |      ________   1 + tan (x)|     \     \/ tan(x) /                                                      \                  tan   (x) /     \      \/ tan(x) / \     \/ tan(x) /                 |     |            2      |       ________|                 |  /           2   \     |            2      |       ________|                 |  /            2   \     /           2   \                   |            2      |       ________|                 |     /            2   \ /           2   \ |            2      |       ________|                 |                                  /                        2   \ |            2      |       ________|                 |
                   |                                                                      |    1 + tan (x)|  |     1 + tan (x)|     |    1 + tan (x)|  /      ________\   /       2   \ /      ________\ |      3/2      4*\1 + tan (x)/   3*\1 + tan (x)/ |     /       2   \ |     1 + tan (x)| |      ________   1 + tan (x)|     /       2   \ |    1 + tan (x)| |      ________   1 + tan (x)|     /       2   \ |    1 + tan (x)| /      ________\ |      ________   1 + tan (x)||     |                   |    1 + tan (x)| /      ________\| |                                                 |    1 + tan (x)|  /      ________\     |     1 + tan (x)| |    1 + tan (x)|   /       2   \ /      ________\ |      ________   1 + tan (x)||                                                |                   |    1 + tan (x)| /      ________\|   2*|-2 + ----------- + ----------------------------------|*||-2 + -----------|  + \1 + tan (x)/*\x - \/ tan(x) /*|- 4*\/ tan(x)  + -----------| + -------------------------------------- + -------------------------------------------------------------- + -------------------------------------------------------|     |     1 + tan (x)   \     \/ tan(x) /                 |  |    1 + tan (x)|     |     1 + tan (x)   \     \/ tan(x) /                 |  |     1 + tan (x)|     |    1 + tan (x)|  /      ________\ |     1 + tan (x)   \     \/ tan(x) /                 |     |     1 + tan (x)| |    1 + tan (x)| |     1 + tan (x)   \     \/ tan(x) /                 |   /       2   \ /      ________\ |      ________   1 + tan (x)| |     1 + tan (x)   \     \/ tan(x) /                 |
                   |              /                                                2\   6*|2 + -----------| *|-2 + -----------|   6*|2 + -----------| *\x - \/ tan(x) /   \1 + tan (x)/*\x - \/ tan(x) /*|16*tan   (x) - --------------- + ----------------|   3*\1 + tan (x)/*|-2 + -----------|*|- 4*\/ tan(x)  + -----------|   3*\1 + tan (x)/*|2 + -----------|*|- 4*\/ tan(x)  + -----------|   6*\1 + tan (x)/*|2 + -----------|*\x - \/ tan(x) /*|- 4*\/ tan(x)  + -----------||     |                   |2 + -----------|*\x - \/ tan(x) /| |                                               2*|2 + -----------| *\x - \/ tan(x) /   2*|-2 + -----------|*|2 + -----------|   \1 + tan (x)/*\x - \/ tan(x) /*|- 4*\/ tan(x)  + -----------||                                                |                   |2 + -----------|*\x - \/ tan(x) /|     |        ________                   ________          | ||        ________|                                   |                     3/2    |                             2                                            ________                                                      ________                    |   2*|-2 + ----------- + ----------------------------------| *|2 + -----------|   2*|-2 + ----------- + ----------------------------------| *|-2 + -----------|   2*|2 + -----------| *\x - \/ tan(x) /*|-2 + ----------- + ----------------------------------|   2*|-2 + -----------|*|2 + -----------|*|-2 + ----------- + ----------------------------------|   \1 + tan (x)/*\x - \/ tan(x) /*|- 4*\/ tan(x)  + -----------|*|-2 + ----------- + ----------------------------------|
                   |              |                 /       2   \     /       2   \ |     |       ________|  |        ________|     |       ________|                                                    |                    ________           5/2       |                   |        ________| |                     3/2    |                   |       ________| |                     3/2    |                   |       ________|                  |                     3/2    ||     |            2      |       ________|                 | |              /                        2   \     |       ________|                       |        ________| |       ________|                                  |                     3/2    ||                 /                        2   \ |            2      |       ________|                 |     \      \/ tan(x)              x + \/ tan(x)           / |\      \/ tan(x) /                                   \                  tan   (x) /             /      ________\                                       x + \/ tan(x)                                                 x + \/ tan(x)                     |     |        ________                   ________          |  |       ________|     |        ________                   ________          |  |        ________|     |       ________|                   |        ________                   ________          |     |        ________| |       ________| |        ________                   ________          |                                  |                     3/2    | |        ________                   ________          |
  /      ________\ |/       2   \ |      3/2      4*\1 + tan (x)/   3*\1 + tan (x)/ |     \     \/ tan(x) /  \      \/ tan(x) /     \     \/ tan(x) /                                                    \                  \/ tan(x)         tan   (x)    /                   \      \/ tan(x) / \                  tan   (x) /                   \     \/ tan(x) / \                  tan   (x) /                   \     \/ tan(x) /                  \                  tan   (x) /|     |     1 + tan (x)   \     \/ tan(x) /                 | |/       2   \ |      ________   1 + tan (x)|     \     \/ tan(x) /                       \      \/ tan(x) / \     \/ tan(x) /                                  \                  tan   (x) /|   /       2   \ |      ________   1 + tan (x)| |     1 + tan (x)   \     \/ tan(x) /                 |                                                             \                                                                                                \x + \/ tan(x) /                                                                                                                                       /     \      \/ tan(x)              x + \/ tan(x)           /  \     \/ tan(x) /     \      \/ tan(x)              x + \/ tan(x)           /  \      \/ tan(x) /     \     \/ tan(x) /                   \      \/ tan(x)              x + \/ tan(x)           /     \      \/ tan(x) / \     \/ tan(x) / \      \/ tan(x)              x + \/ tan(x)           /                                  \                  tan   (x) / \      \/ tan(x)              x + \/ tan(x)           /
- \x - \/ tan(x) /*|\1 + tan (x)/*|16*tan   (x) - --------------- + ----------------| + --------------------------------------- + ------------------------------------- + ---------------------------------------------------------------------------------- + ----------------------------------------------------------------- + ---------------------------------------------------------------- + ---------------------------------------------------------------------------------| - 2*|-2 + ----------- + ----------------------------------|*|\1 + tan (x)/*|- 4*\/ tan(x)  + -----------| + ------------------------------------- + -------------------------------------- + -------------------------------------------------------------| + \1 + tan (x)/*|- 4*\/ tan(x)  + -----------|*|-2 + ----------- + ----------------------------------| - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------- + ----------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------
                   |              |                    ________           5/2       |                              2                                        3                                                     ________                                                                     ________                                                           ________                                                                            2                                |     |        ________                   ________          | |              |                     3/2    |                             2                               ________                                             ________                       |                 |                     3/2    | |        ________                   ________          |                                                                                                                                                            ________                                                                                                                                                                                          ________                                                                        ________                                                                                        2                                                                                       ________                                                                                                     ________                                                   
                   |              \                  \/ tan(x)         tan   (x)    /              /      ________\                         /      ________\                                                x + \/ tan(x)                                                                x + \/ tan(x)                                                      x + \/ tan(x)                                                             /      ________\                                 |     \      \/ tan(x)              x + \/ tan(x)           / |              \                  tan   (x) /             /      ________\                          x + \/ tan(x)                                        x + \/ tan(x)                        |                 \                  tan   (x) / \      \/ tan(x)              x + \/ tan(x)           /                                                                                                                                                      x - \/ tan(x)                                                                                                                                                                                     x + \/ tan(x)                                                                   x - \/ tan(x)                                                                         /      ________\                                                                                  x + \/ tan(x)                                                                                                x + \/ tan(x)                                                    
                   \                                                                               \x + \/ tan(x) /                         \x + \/ tan(x) /                                                                                                                                                                                                                                                                          \x + \/ tan(x) /                                 /                                                             \                                                         \x + \/ tan(x) /                                                                                                                    /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     \x + \/ tan(x) /                                                                                                                                                                                                                                                                
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             /      ________\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           4*\x + \/ tan(x) /

$$\frac{- \left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{\frac{5}{2}}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)}{\sqrt{\tan{\left(x \right)}}} + 16 \tan^{\frac{3}{2}}{\left(x \right)}\right)}{x + \sqrt{\tan{\left(x \right)}}} + \frac{6 \left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{3}{2}}{\left(x \right)}} - 4 \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(x + \sqrt{\tan{\left(x \right)}}\right)^{2}} + \frac{6 \left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right)^{3}}{\left(x + \sqrt{\tan{\left(x \right)}}\right)^{3}} + \left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{\frac{5}{2}}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)}{\sqrt{\tan{\left(x \right)}}} + 16 \tan^{\frac{3}{2}}{\left(x \right)}\right) + \frac{3 \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{3}{2}}{\left(x \right)}} - 4 \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} - 2\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{x + \sqrt{\tan{\left(x \right)}}} + \frac{3 \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{3}{2}}{\left(x \right)}} - 4 \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{x + \sqrt{\tan{\left(x \right)}}} + \frac{6 \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} - 2\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right)^{2}}{\left(x + \sqrt{\tan{\left(x \right)}}\right)^{2}}\right) + \frac{\left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{3}{2}}{\left(x \right)}} - 4 \sqrt{\tan{\left(x \right)}}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{\left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right)}{x + \sqrt{\tan{\left(x \right)}}} + \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} - 2\right)}{x + \sqrt{\tan{\left(x \right)}}} + \frac{2 \left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right)^{2} \left(\frac{\left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right)}{x + \sqrt{\tan{\left(x \right)}}} + \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} - 2\right)}{\left(x + \sqrt{\tan{\left(x \right)}}\right)^{2}} + \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{3}{2}}{\left(x \right)}} - 4 \sqrt{\tan{\left(x \right)}}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{\left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right)}{x + \sqrt{\tan{\left(x \right)}}} + \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} - 2\right) - 2 \left(\frac{\left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right)}{x + \sqrt{\tan{\left(x \right)}}} + \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} - 2\right) \left(\frac{\left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{3}{2}}{\left(x \right)}} - 4 \sqrt{\tan{\left(x \right)}}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{x + \sqrt{\tan{\left(x \right)}}} + \frac{2 \left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right)^{2}}{\left(x + \sqrt{\tan{\left(x \right)}}\right)^{2}} + \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{3}{2}}{\left(x \right)}} - 4 \sqrt{\tan{\left(x \right)}}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \frac{2 \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} - 2\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right)}{x + \sqrt{\tan{\left(x \right)}}}\right) + \frac{2 \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} - 2\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right) \left(\frac{\left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right)}{x + \sqrt{\tan{\left(x \right)}}} + \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} - 2\right)}{x + \sqrt{\tan{\left(x \right)}}} + \frac{2 \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right) \left(\frac{\left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right)}{x + \sqrt{\tan{\left(x \right)}}} + \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} - 2\right)^{2}}{x + \sqrt{\tan{\left(x \right)}}} + \frac{2 \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} - 2\right) \left(\frac{\left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right)}{x + \sqrt{\tan{\left(x \right)}}} + \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} - 2\right)^{2}}{x - \sqrt{\tan{\left(x \right)}}} - \frac{2 \left(\frac{\left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right)}{x + \sqrt{\tan{\left(x \right)}}} + \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} - 2\right) \left(\frac{\left(x - \sqrt{\tan{\left(x \right)}}\right)^{2} \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{3}{2}}{\left(x \right)}} - 4 \sqrt{\tan{\left(x \right)}}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{x + \sqrt{\tan{\left(x \right)}}} + \frac{3 \left(x - \sqrt{\tan{\left(x \right)}}\right)^{2} \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right)^{2}}{\left(x + \sqrt{\tan{\left(x \right)}}\right)^{2}} + \left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{3}{2}}{\left(x \right)}} - 4 \sqrt{\tan{\left(x \right)}}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \frac{4 \left(x - \sqrt{\tan{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} - 2\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} + 2\right)}{x + \sqrt{\tan{\left(x \right)}}} + \left(\frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{\tan{\left(x \right)}}} - 2\right)^{2}\right)}{x - \sqrt{\tan{\left(x \right)}}}}{4 \left(x + \sqrt{\tan{\left(x \right)}}\right)^{2}}$$

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