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((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

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

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src] 
                2
/      ________\ 
|x - \/ tan(x) | 
|--------------| 
|      ________| 
\x + \/ tan(x) /
(x−tan⁡(x)x+tan⁡(x))2\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

  1. Sustituimos u=x−tan⁡(x)x+tan⁡(x)u = \frac{x - \sqrt{\tan{\left(x \right)}}}{x + \sqrt{\tan{\left(x \right)}}}.

  2. Según el principio, aplicamos: u2u^{2} tenemos 2u2 u

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddxx−tan⁡(x)x+tan⁡(x)\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:

      ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=x−tan⁡(x)f{\left(x \right)} = x - \sqrt{\tan{\left(x \right)}} y g(x)=x+tan⁡(x)g{\left(x \right)} = x + \sqrt{\tan{\left(x \right)}}.

      Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. diferenciamos x−tan⁡(x)x - \sqrt{\tan{\left(x \right)}} miembro por miembro:

        1. Según el principio, aplicamos: xx tenemos 11

        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⁡(x)u = \tan{\left(x \right)}.

          2. Según el principio, aplicamos: u\sqrt{u} tenemos 12u\frac{1}{2 \sqrt{u}}

          3. Luego se aplica una cadena de reglas. Multiplicamos por ddxtan⁡(x)\frac{d}{d x} \tan{\left(x \right)}:

            1. Reescribimos las funciones para diferenciar:

              tan⁡(x)=sin⁡(x)cos⁡(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

            2. Se aplica la regla de la derivada parcial:

              ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} y g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}.

              Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

              1. La derivada del seno es igual al coseno:

                ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

              Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

              1. La derivada del coseno es igual a menos el seno:

                ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

              Ahora aplicamos la regla de la derivada de una divesión:

              sin⁡2(x)+cos⁡2(x)cos⁡2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

            Como resultado de la secuencia de reglas:

            sin⁡2(x)+cos⁡2(x)2cos⁡2(x)tan⁡(x)\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: −sin⁡2(x)+cos⁡2(x)2cos⁡2(x)tan⁡(x)- \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: −sin⁡2(x)+cos⁡2(x)2cos⁡2(x)tan⁡(x)+1- \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 ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. diferenciamos x+tan⁡(x)x + \sqrt{\tan{\left(x \right)}} miembro por miembro:

        1. Según el principio, aplicamos: xx tenemos 11

        2. Sustituimos u=tan⁡(x)u = \tan{\left(x \right)}.

        3. Según el principio, aplicamos: u\sqrt{u} tenemos 12u\frac{1}{2 \sqrt{u}}

        4. Luego se aplica una cadena de reglas. Multiplicamos por ddxtan⁡(x)\frac{d}{d x} \tan{\left(x \right)}:

          1. ddxtan⁡(x)=1cos⁡2(x)\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}

          Como resultado de la secuencia de reglas:

          sin⁡2(x)+cos⁡2(x)2cos⁡2(x)tan⁡(x)\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: sin⁡2(x)+cos⁡2(x)2cos⁡2(x)tan⁡(x)+1\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:

      −(x−tan⁡(x))(sin⁡2(x)+cos⁡2(x)2cos⁡2(x)tan⁡(x)+1)+(x+tan⁡(x))(−sin⁡2(x)+cos⁡2(x)2cos⁡2(x)tan⁡(x)+1)(x+tan⁡(x))2\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:

    2(x−tan⁡(x))(−(x−tan⁡(x))(sin⁡2(x)+cos⁡2(x)2cos⁡2(x)tan⁡(x)+1)+(x+tan⁡(x))(−sin⁡2(x)+cos⁡2(x)2cos⁡2(x)tan⁡(x)+1))(x+tan⁡(x))3\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}}

  4. Simplificamos:

    −2(x−sin⁡(2x))(x−tan⁡(x))(x+tan⁡(x))3cos⁡2(x)tan⁡(x)- \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:

−2(x−sin⁡(2x))(x−tan⁡(x))(x+tan⁡(x))3cos⁡2(x)tan⁡(x)- \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
02468-8-6-4-2-1010-100000100000
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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)
(x−tan⁡(x))2(x+tan⁡(x))2(x+tan⁡(x))(2(x−tan⁡(x))(−tan⁡2(x)2+12tan⁡(x)−1)(x+tan⁡(x))2+2(−tan⁡2(x)2+12tan⁡(x)+1)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) /
(x−tan⁡(x))((x−tan⁡(x))(tan⁡2(x)+1tan⁡32(x)−4tan⁡(x))(tan⁡2(x)+1)x+tan⁡(x)+2(x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)2(x+tan⁡(x))2+(tan⁡2(x)+1tan⁡32(x)−4tan⁡(x))(tan⁡2(x)+1)+2(tan⁡2(x)+1tan⁡(x)−2)(tan⁡2(x)+1tan⁡(x)+2)x+tan⁡(x))−(x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)((x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)x+tan⁡(x)+tan⁡2(x)+1tan⁡(x)−2)x+tan⁡(x)−(tan⁡2(x)+1tan⁡(x)−2)((x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)x+tan⁡(x)+tan⁡2(x)+1tan⁡(x)−2)+2((x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)x+tan⁡(x)+tan⁡2(x)+1tan⁡(x)−2)22(x+tan⁡(x))2\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) /
−(x−tan⁡(x))((x−tan⁡(x))(tan⁡2(x)+1)(3(tan⁡2(x)+1)2tan⁡52(x)−4(tan⁡2(x)+1)tan⁡(x)+16tan⁡32(x))x+tan⁡(x)+6(x−tan⁡(x))(tan⁡2(x)+1tan⁡32(x)−4tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)(tan⁡2(x)+1)(x+tan⁡(x))2+6(x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)3(x+tan⁡(x))3+(tan⁡2(x)+1)(3(tan⁡2(x)+1)2tan⁡52(x)−4(tan⁡2(x)+1)tan⁡(x)+16tan⁡32(x))+3(tan⁡2(x)+1tan⁡32(x)−4tan⁡(x))(tan⁡2(x)+1tan⁡(x)−2)(tan⁡2(x)+1)x+tan⁡(x)+3(tan⁡2(x)+1tan⁡32(x)−4tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)(tan⁡2(x)+1)x+tan⁡(x)+6(tan⁡2(x)+1tan⁡(x)−2)(tan⁡2(x)+1tan⁡(x)+2)2(x+tan⁡(x))2)+(x−tan⁡(x))(tan⁡2(x)+1tan⁡32(x)−4tan⁡(x))(tan⁡2(x)+1)((x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)x+tan⁡(x)+tan⁡2(x)+1tan⁡(x)−2)x+tan⁡(x)+2(x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)2((x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)x+tan⁡(x)+tan⁡2(x)+1tan⁡(x)−2)(x+tan⁡(x))2+(tan⁡2(x)+1tan⁡32(x)−4tan⁡(x))(tan⁡2(x)+1)((x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)x+tan⁡(x)+tan⁡2(x)+1tan⁡(x)−2)−2((x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)x+tan⁡(x)+tan⁡2(x)+1tan⁡(x)−2)((x−tan⁡(x))(tan⁡2(x)+1tan⁡32(x)−4tan⁡(x))(tan⁡2(x)+1)x+tan⁡(x)+2(x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)2(x+tan⁡(x))2+(tan⁡2(x)+1tan⁡32(x)−4tan⁡(x))(tan⁡2(x)+1)+2(tan⁡2(x)+1tan⁡(x)−2)(tan⁡2(x)+1tan⁡(x)+2)x+tan⁡(x))+2(tan⁡2(x)+1tan⁡(x)−2)(tan⁡2(x)+1tan⁡(x)+2)((x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)x+tan⁡(x)+tan⁡2(x)+1tan⁡(x)−2)x+tan⁡(x)+2(tan⁡2(x)+1tan⁡(x)+2)((x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)x+tan⁡(x)+tan⁡2(x)+1tan⁡(x)−2)2x+tan⁡(x)+2(tan⁡2(x)+1tan⁡(x)−2)((x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)x+tan⁡(x)+tan⁡2(x)+1tan⁡(x)−2)2x−tan⁡(x)−2((x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)+2)x+tan⁡(x)+tan⁡2(x)+1tan⁡(x)−2)((x−tan⁡(x))2(tan⁡2(x)+1tan⁡32(x)−4tan⁡(x))(tan⁡2(x)+1)x+tan⁡(x)+3(x−tan⁡(x))2(tan⁡2(x)+1tan⁡(x)+2)2(x+tan⁡(x))2+(x−tan⁡(x))(tan⁡2(x)+1tan⁡32(x)−4tan⁡(x))(tan⁡2(x)+1)+4(x−tan⁡(x))(tan⁡2(x)+1tan⁡(x)−2)(tan⁡2(x)+1tan⁡(x)+2)x+tan⁡(x)+(tan⁡2(x)+1tan⁡(x)−2)2)x−tan⁡(x)4(x+tan⁡(x))2\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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Gráfico
Derivada de ((x-tgx^1/2)/(x+tgx^1/2))^2
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