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(xln^ tres (x)sqrt(x^ cinco + nueve))/(ctgx(2x^ cuatro + cinco)^ cinco)
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(xln en el grado tres (x) raíz cuadrada de (x en el grado cinco más nueve)) dividir por (ctgx(2x en el grado cuatro más cinco) en el grado cinco)
(xln^3(x)√(x^5+9))/(ctgx(2x^4+5)^5)
(xln3(x)sqrt(x5+9))/(ctgx(2x4+5)5)
xln3xsqrtx5+9/ctgx2x4+55
(xln³(x)sqrt(x⁵+9))/(ctgx(2x⁴+5)⁵)
(xln en el grado 3(x)sqrt(x en el grado 5+9))/(ctgx(2x en el grado 4+5) en el grado 5)
xln^3xsqrtx^5+9/ctgx2x^4+5^5
(xln^3(x)sqrt(x^5+9)) dividir por (ctgx(2x^4+5)^5)
Expresiones semejantes
(xln^3(x)sqrt(x^5+9))/(ctgx(2x^4-5)^5)
(xln^3(x)sqrt(x^5-9))/(ctgx(2x^4+5)^5)

Derivada de la función/ (xln^3(x)sqrt(x^5+9))/(ctgx(2x^4+5)^5)

Derivada de (xln^3(x)sqrt(x^5+9))/(ctgx(2x^4+5)^5)

Solución

Ha introducido [src]

             ________
     3      /  5     
x*log (x)*\/  x  + 9 
---------------------
                   5 
         /   4    \  
  cot(x)*\2*x  + 5/

$$\frac{x \log{\left(x \right)}^{3} \sqrt{x^{5} + 9}}{\left(2 x^{4} + 5\right)^{5} \cot{\left(x \right)}}$$

((x*log(x)^3)*sqrt(x^5 + 9))/((cot(x)*(2*x^4 + 5)^5))

Solución detallada

Se aplica la regla de la derivada parcial:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x \sqrt{x^{5} + 9} \log{\left(x \right)}^{3}$ y $g{\left(x \right)} = \left(2 x^{4} + 5\right)^{5} \cot{\left(x \right)}$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} h{\left(x \right)} = f{\left(x \right)} g{\left(x \right)} \frac{d}{d x} h{\left(x \right)} + f{\left(x \right)} h{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} h{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Según el principio, aplicamos: $x$ tenemos $1$
  $g{\left(x \right)} = \sqrt{x^{5} + 9}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Sustituimos $u = x^{5} + 9$ .
  2. Según el principio, aplicamos: $\sqrt{u}$ tenemos $\frac{1}{2 \sqrt{u}}$
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \left(x^{5} + 9\right)$ :
    1. diferenciamos $x^{5} + 9$ miembro por miembro:
      1. La derivada de una constante $9$ es igual a cero.
      2. Según el principio, aplicamos: $x^{5}$ tenemos $5 x^{4}$
      Como resultado de: $5 x^{4}$
    Como resultado de la secuencia de reglas:
    
    $\frac{5 x^{4}}{2 \sqrt{x^{5} + 9}}$
  $h{\left(x \right)} = \log{\left(x \right)}^{3}$ ; calculamos $\frac{d}{d x} h{\left(x \right)}$ :
  1. Sustituimos $u = \log{\left(x \right)}$ .
  2. Según el principio, aplicamos: $u^{3}$ tenemos $3 u^{2}$
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \log{\left(x \right)}$ :
    1. Derivado $\log{\left(x \right)}$ es $\frac{1}{x}$ .
    Como resultado de la secuencia de reglas:
    
    $\frac{3 \log{\left(x \right)}^{2}}{x}$
  Como resultado de: $\frac{5 x^{5} \log{\left(x \right)}^{3}}{2 \sqrt{x^{5} + 9}} + \sqrt{x^{5} + 9} \log{\left(x \right)}^{3} + 3 \sqrt{x^{5} + 9} \log{\left(x \right)}^{2}$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \left(2 x^{4} + 5\right)^{5}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Sustituimos $u = 2 x^{4} + 5$ .
  2. Según el principio, aplicamos: $u^{5}$ tenemos $5 u^{4}$
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \left(2 x^{4} + 5\right)$ :
    1. diferenciamos $2 x^{4} + 5$ miembro por miembro:
      1. La derivada de una constante $5$ es igual a cero.
      2. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
        
        Según el principio, aplicamos: $x^{4}$ tenemos $4 x^{3}$
        
        Entonces, como resultado: $8 x^{3}$
      Como resultado de: $8 x^{3}$
    Como resultado de la secuencia de reglas:
    
    $40 x^{3} \left(2 x^{4} + 5\right)^{4}$
  $g{\left(x \right)} = \cot{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Hay varias formas de calcular esta derivada.
    Method #1
    1. Reescribimos las funciones para diferenciar:
      
      $\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}$
    2. Sustituimos $u = \tan{\left(x \right)}$ .
    3. Según el principio, aplicamos: $\frac{1}{u}$ tenemos $- \frac{1}{u^{2}}$
    4. 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:
      
      $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
    Method #2
    1. Reescribimos las funciones para diferenciar:
      
      $\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
    2. Se aplica la regla de la derivada parcial:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(x \right)}$ y $g{\left(x \right)} = \sin{\left(x \right)}$ .
      
      Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
      
      La derivada del coseno es igual a menos el seno:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      
      Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
      
      La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      
      Ahora aplicamos la regla de la derivada de una divesión:
      
      $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
  Como resultado de: $40 x^{3} \left(2 x^{4} + 5\right)^{4} \cot{\left(x \right)} - \frac{\left(2 x^{4} + 5\right)^{5} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
Ahora aplicamos la regla de la derivada de una divesión:

$\frac{- x \sqrt{x^{5} + 9} \left(40 x^{3} \left(2 x^{4} + 5\right)^{4} \cot{\left(x \right)} - \frac{\left(2 x^{4} + 5\right)^{5} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}\right) \log{\left(x \right)}^{3} + \left(2 x^{4} + 5\right)^{5} \left(\frac{5 x^{5} \log{\left(x \right)}^{3}}{2 \sqrt{x^{5} + 9}} + \sqrt{x^{5} + 9} \log{\left(x \right)}^{3} + 3 \sqrt{x^{5} + 9} \log{\left(x \right)}^{2}\right) \cot{\left(x \right)}}{\left(2 x^{4} + 5\right)^{10} \cot^{2}{\left(x \right)}}$
Simplificamos:

$\frac{\left(- x \left(x^{5} + 9\right) \left(- 2 x^{4} + 20 x^{3} \sin{\left(2 x \right)} - 5\right) \log{\left(x \right)} + \frac{\left(2 x^{4} + 5\right) \left(5 x^{5} \log{\left(x \right)} + 2 \left(x^{5} + 9\right) \left(\log{\left(x \right)} + 3\right)\right) \sin{\left(2 x \right)}}{4}\right) \log{\left(x \right)}^{2}}{\left(2 x^{4} + 5\right)^{6} \sqrt{x^{5} + 9} \cos^{2}{\left(x \right)}}$

Respuesta:

$\frac{\left(- x \left(x^{5} + 9\right) \left(- 2 x^{4} + 20 x^{3} \sin{\left(2 x \right)} - 5\right) \log{\left(x \right)} + \frac{\left(2 x^{4} + 5\right) \left(5 x^{5} \log{\left(x \right)} + 2 \left(x^{5} + 9\right) \left(\log{\left(x \right)} + 3\right)\right) \sin{\left(2 x \right)}}{4}\right) \log{\left(x \right)}^{2}}{\left(2 x^{4} + 5\right)^{6} \sqrt{x^{5} + 9} \cos^{2}{\left(x \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

                                                                              ________         /            5                                  4       \
                   /   ________                             5    3   \       /  5         3    |  /   4    \  /        2   \       3 /   4    \        |
        1          |  /  5      /   3           2   \    5*x *log (x)|   x*\/  x  + 9 *log (x)*\- \2*x  + 5/ *\-1 - cot (x)/ - 40*x *\2*x  + 5/ *cot(x)/
------------------*|\/  x  + 9 *\log (x) + 3*log (x)/ + -------------| + -------------------------------------------------------------------------------
          5        |                                         ________|                                           10                                     
/   4    \         |                                        /  5     |                                 /   4    \      2                                
\2*x  + 5/ *cot(x) \                                    2*\/  x  + 9 /                                 \2*x  + 5/  *cot (x)

$$\frac{x \sqrt{x^{5} + 9} \left(- 40 x^{3} \left(2 x^{4} + 5\right)^{4} \cot{\left(x \right)} - \left(2 x^{4} + 5\right)^{5} \left(- \cot^{2}{\left(x \right)} - 1\right)\right) \log{\left(x \right)}^{3}}{\left(2 x^{4} + 5\right)^{10} \cot^{2}{\left(x \right)}} + \frac{1}{\left(2 x^{4} + 5\right)^{5} \cot{\left(x \right)}} \left(\frac{5 x^{5} \log{\left(x \right)}^{3}}{2 \sqrt{x^{5} + 9}} + \sqrt{x^{5} + 9} \left(\log{\left(x \right)}^{3} + 3 \log{\left(x \right)}^{2}\right)\right)$$

Simplificar

Segunda derivada [src]

/                                                                                                                                 /     ________                   5       \                                /                                                                           /                          2                                                                                \                                                                                                                \\       
|                                                                     /         5 \   /  /       2   \ /       4\       3       \ |    /      5                 5*x *log(x)|               ________         |/         2           3  \                                                 |     6          /       4\  /       2   \              3 /       2   \ /       4\       2 /       4\       |   /       2   \ /  /       2   \ /       4\       3       \       3 /  /       2   \ /       4\       3       \||       
|                                                           4    2    |      5*x  |   \- \1 + cot (x)/*\5 + 2*x / + 40*x *cot(x)/*|2*\/  9 + x  *(3 + log(x)) + -----------|*log(x)       /      5     2    ||  1 + cot (x)    40*x   | /  /       2   \ /       4\       3       \   2*\640*x *cot(x) + \5 + 2*x / *\1 + cot (x)/*cot(x) - 40*x *\1 + cot (x)/*\5 + 2*x / + 60*x *\5 + 2*x /*cot(x)/   \1 + cot (x)/*\- \1 + cot (x)/*\5 + 2*x / + 40*x *cot(x)/   40*x *\- \1 + cot (x)/*\5 + 2*x / + 40*x *cot(x)/||       
|     ________                                           5*x *log (x)*|-8 + ------|                                               |                                ________|          x*\/  9 + x  *log (x)*||- ----------- + --------|*\- \1 + cot (x)/*\5 + 2*x / + 40*x *cot(x)/ - --------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------- + -------------------------------------------------||       
|    /      5                    4                                    |          5|                                               |                               /      5 |                                ||     cot(x)            4|                                                                                                          4                                                                                cot(x)                                                        4                    ||       
|3*\/  9 + x  *(2 + log(x))   5*x *(3 + log(x))*log(x)                \     9 + x /                                               \                             \/  9 + x  /                                \\                5 + 2*x /                                                                                                   5 + 2*x                                                                                                                                        5 + 2*x                     /|       
|-------------------------- + ------------------------ - -------------------------- - --------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*log(x)
|            x                         ________                     ________                                                /       4\                                                                                                                                                                                                       /       4\                                                                                                                                                               |       
|                                     /      5                     /      5                                                 \5 + 2*x /*cot(x)                                                                                                                                                                                                \5 + 2*x /*cot(x)                                                                                                                                                        |       
\                                   \/  9 + x                  4*\/  9 + x                                                                                                                                                                                                                                                                                                                                                                                                                                            /       
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                5                                                                                                                                                                                                                                                             
                                                                                                                                                                                                                                                      /       4\                                                                                                                                                                                                                                                              
                                                                                                                                                                                                                                                      \5 + 2*x / *cot(x)

$$\frac{\left(- \frac{5 x^{4} \left(\frac{5 x^{5}}{x^{5} + 9} - 8\right) \log{\left(x \right)}^{2}}{4 \sqrt{x^{5} + 9}} + \frac{5 x^{4} \left(\log{\left(x \right)} + 3\right) \log{\left(x \right)}}{\sqrt{x^{5} + 9}} + \frac{x \sqrt{x^{5} + 9} \left(\frac{40 x^{3} \left(40 x^{3} \cot{\left(x \right)} - \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right)}{2 x^{4} + 5} + \left(\frac{40 x^{3}}{2 x^{4} + 5} - \frac{\cot^{2}{\left(x \right)} + 1}{\cot{\left(x \right)}}\right) \left(40 x^{3} \cot{\left(x \right)} - \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right) - \frac{\left(40 x^{3} \cot{\left(x \right)} - \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot{\left(x \right)}} - \frac{2 \left(640 x^{6} \cot{\left(x \right)} - 40 x^{3} \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right) + 60 x^{2} \left(2 x^{4} + 5\right) \cot{\left(x \right)} + \left(2 x^{4} + 5\right)^{2} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right)}{2 x^{4} + 5}\right) \log{\left(x \right)}^{2}}{\left(2 x^{4} + 5\right) \cot{\left(x \right)}} - \frac{\left(40 x^{3} \cot{\left(x \right)} - \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right) \left(\frac{5 x^{5} \log{\left(x \right)}}{\sqrt{x^{5} + 9}} + 2 \sqrt{x^{5} + 9} \left(\log{\left(x \right)} + 3\right)\right) \log{\left(x \right)}}{\left(2 x^{4} + 5\right) \cot{\left(x \right)}} + \frac{3 \sqrt{x^{5} + 9} \left(\log{\left(x \right)} + 2\right)}{x}\right) \log{\left(x \right)}}{\left(2 x^{4} + 5\right)^{5} \cot{\left(x \right)}}$$

Simplificar

Tercera derivada [src]

                                                                                                                                                                                                                                      /                                           /         5 \                            \                                                                                                                                                                                                                                                                                                                                                                                                  /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       /         2           3  \ /                          2                                                                                \                                                                                                                                                                                             /         2           3  \                                                     /         2           3  \                                                                                                                                                                                                                                               \
                                                                                                                                                                                                                                      |                                 4    2    |      5*x  |                            |                                                                                                                                                                                                                                                                                                                                                                                                  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |  1 + cot (x)    40*x   | |     6          /       4\  /       2   \              3 /       2   \ /       4\       2 /       4\       |                                                                                                                                                                               /       2   \ |  1 + cot (x)    40*x   | /  /       2   \ /       4\       3       \       3 |  1 + cot (x)    40*x   | /  /       2   \ /       4\       3       \                                                                                                                                                                                                   |
                                                                                                                                                                                                                                      |      ________                5*x *log (x)*|-8 + ------|                            |                                                               /                                                                           /                          2                                                                                \                                                                                                                \                         |    /               2           3                                                                         2                             3                                         2                                                       2                     \                                                                                                               /                           2                                               \                                                                        2                                               2*|- ----------- + --------|*\640*x *cot(x) + \5 + 2*x / *\1 + cot (x)/*cot(x) - 40*x *\1 + cot (x)/*\5 + 2*x / + 60*x *\5 + 2*x /*cot(x)/                                                               /                          2                                                                                \   \1 + cot (x)/*|- ----------- + --------|*\- \1 + cot (x)/*\5 + 2*x / + 40*x *cot(x)/   40*x *|- ----------- + --------|*\- \1 + cot (x)/*\5 + 2*x / + 40*x *cot(x)/                   /                          2                                                                                \                                                                   |
                                                                                     /         5          10 \                                                                                                                        |     /      5                              |          5|       4                    |                    /     ________                   5       \ |/         2           3  \                                                 |     6          /       4\  /       2   \              3 /       2   \ /       4\       2 /       4\       |   /       2   \ /  /       2   \ /       4\       3       \       3 /  /       2   \ /       4\       3       \|        ________         |    |  /       2   \  /       4\           9                6 /       2   \ /       4\        2 /       4\  /       2   \     /       4\     2    /       2   \         /       4\                 5 /       4\               3 /       4\  /       2   \       |                                                                                                               |              /       2   \            6          2         3 /       2   \|         6 /  /       2   \ /       4\       3       \     /       2   \  /  /       2   \ /       4\       3       \     |     cot(x)            4|                                                                                                                      2 /  /       2   \ /       4\       3       \        3 |     6          /       4\  /       2   \              3 /       2   \ /       4\       2 /       4\       |                 |     cot(x)            4|                                                     |     cot(x)            4|                                                 /       2   \ |     6          /       4\  /       2   \              3 /       2   \ /       4\       2 /       4\       |        3 /       2   \ /  /       2   \ /       4\       3       \|
                                                                           3    3    |     40*x       25*x   |                                             /         5 \                  /  /       2   \ /       4\       3       \ |12*\/  9 + x  *(2 + log(x))                \     9 + x /   20*x *(3 + log(x))*log(x)|               2    |    /      5                 5*x *log(x)| ||  1 + cot (x)    40*x   | /  /       2   \ /       4\       3       \   2*\640*x *cot(x) + \5 + 2*x / *\1 + cot (x)/*cot(x) - 40*x *\1 + cot (x)/*\5 + 2*x / + 60*x *\5 + 2*x /*cot(x)/   \1 + cot (x)/*\- \1 + cot (x)/*\5 + 2*x / + 40*x *cot(x)/   40*x *\- \1 + cot (x)/*\5 + 2*x / + 40*x *cot(x)/|       /      5     3    |  2*\- \1 + cot (x)/ *\5 + 2*x /  + 15360*x *cot(x) - 1920*x *\1 + cot (x)/*\5 + 2*x / - 180*x *\5 + 2*x / *\1 + cot (x)/ - 2*\5 + 2*x / *cot (x)*\1 + cot (x)/ + 120*x*\5 + 2*x / *cot(x) + 5760*x *\5 + 2*x /*cot(x) + 120*x *\5 + 2*x / *\1 + cot (x)/*cot(x)/     /       2   \ /  /       2   \ /       4\       3       \     /  /       2   \ /       4\       3       \ |       2      \1 + cot (x)/       960*x       60*x      40*x *\1 + cot (x)/|   3520*x *\- \1 + cot (x)/*\5 + 2*x / + 40*x *cot(x)/   3*\1 + cot (x)/ *\- \1 + cot (x)/*\5 + 2*x / + 40*x *cot(x)/     \                5 + 2*x /                                                                                                                 120*x *\- \1 + cot (x)/*\5 + 2*x / + 40*x *cot(x)/   240*x *\640*x *cot(x) + \5 + 2*x / *\1 + cot (x)/*cot(x) - 40*x *\1 + cot (x)/*\5 + 2*x / + 60*x *\5 + 2*x /*cot(x)/                 \                5 + 2*x /                                                     \                5 + 2*x /                                               6*\1 + cot (x)/*\640*x *cot(x) + \5 + 2*x / *\1 + cot (x)/*cot(x) - 40*x *\1 + cot (x)/*\5 + 2*x / + 60*x *\5 + 2*x /*cot(x)/   160*x *\1 + cot (x)/*\- \1 + cot (x)/*\5 + 2*x / + 40*x *cot(x)/|
                                                                       15*x *log (x)*|16 - ------ + ---------|                                   3    2    |      5*x  |                3*\- \1 + cot (x)/*\5 + 2*x / + 40*x *cot(x)/*|--------------------------- - -------------------------- + -------------------------|*log(x)   3*log (x)*|2*\/  9 + x  *(3 + log(x)) + -----------|*||- ----------- + --------|*\- \1 + cot (x)/*\5 + 2*x / + 40*x *cot(x)/ - --------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------- + -------------------------------------------------|   x*\/  9 + x  *log (x)*|- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + 2*\1 + cot (x)/*\- \1 + cot (x)/*\5 + 2*x / + 40*x *cot(x)/ + 2*\- \1 + cot (x)/*\5 + 2*x / + 40*x *cot(x)/*|1 + cot (x) - -------------- - ----------- + -------- + -------------------| - --------------------------------------------------- - ------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------------ + -------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------ - ---------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------|
       ________                                                                      |          5           2|                               15*x *log (x)*|-8 + ------|*(3 + log(x))                                                 |             x                          ________                     ________       |                    |                                ________| ||     cot(x)            4|                                                                                                          4                                                                                cot(x)                                                        4                    |                         |                                                                                                                                      2                                                                                                                                                                                                                                         |                    2                    2          4    /       4\        |                                 2                                                    2                                                                                                         4                                                                                                4                                                                                      2                                                                                               cot(x)                                                                                   4                                                                                           /       4\                                                                                       /       4\                               |
      /      5  /          2                                       \                 |     9 + x    /     5\ |       3                                     |          5|                                                              |                                       /      5                     /      5        |                    |                               /      5 | \\                5 + 2*x /                                                                                                   5 + 2*x                                                                                                                                        5 + 2*x                     /                         |                                                                                                                            /       4\                                                                                                                                                                                                                                          |                 cot (x)       /       4\    5 + 2*x     \5 + 2*x /*cot(x) |                       /       4\                                                  cot (x)                                                                                               5 + 2*x                                                                                          5 + 2*x                                                                             /       4\                                                                                                                                                                                  5 + 2*x                                                                                            \5 + 2*x /*cot(x)                                                                                \5 + 2*x /*cot(x)                        |
  3*\/  9 + x  *\-2 - 2*log (x) + 6*log(x) + 3*(-2 + log(x))*log(x)/                 \              \9 + x / /   45*x *(2 + log(x))*log(x)                 \     9 + x /                                                              \                                     \/  9 + x                    \/  9 + x         /                    \                             \/  9 + x  /                                                                                                                                                                                                                                                                                                                                    \                                                                                                                            \5 + 2*x /                                                                                                                                                                                                                                          \                               \5 + 2*x /                                  /                       \5 + 2*x /                                                                                                                                                                                                                                                                                                                                             \5 + 2*x /                                                                                                                                                                                                                                                                                                                                                                                                                               /
- ------------------------------------------------------------------ + --------------------------------------- + ------------------------- - ---------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                   2                                                     ________                           ________                           ________                                                                               /       4\                                                                                                                                                                                                                                              /       4\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             /       4\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
                                  x                                                     /      5                           /      5                           /      5                                                                              4*\5 + 2*x /*cot(x)                                                                                                                                                                                                                                     2*\5 + 2*x /*cot(x)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      \5 + 2*x /*cot(x)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
                                                                                    8*\/  9 + x                        2*\/  9 + x                        4*\/  9 + x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 /       4\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 \5 + 2*x / *cot(x)

$$\frac{- \frac{15 x^{3} \left(\frac{5 x^{5}}{x^{5} + 9} - 8\right) \left(\log{\left(x \right)} + 3\right) \log{\left(x \right)}^{2}}{4 \sqrt{x^{5} + 9}} + \frac{45 x^{3} \left(\log{\left(x \right)} + 2\right) \log{\left(x \right)}}{2 \sqrt{x^{5} + 9}} + \frac{15 x^{3} \left(\frac{25 x^{10}}{\left(x^{5} + 9\right)^{2}} - \frac{40 x^{5}}{x^{5} + 9} + 16\right) \log{\left(x \right)}^{3}}{8 \sqrt{x^{5} + 9}} + \frac{x \sqrt{x^{5} + 9} \left(- \frac{3520 x^{6} \left(40 x^{3} \cot{\left(x \right)} - \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right)}{\left(2 x^{4} + 5\right)^{2}} - \frac{40 x^{3} \left(\frac{40 x^{3}}{2 x^{4} + 5} - \frac{\cot^{2}{\left(x \right)} + 1}{\cot{\left(x \right)}}\right) \left(40 x^{3} \cot{\left(x \right)} - \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right)}{2 x^{4} + 5} + \frac{160 x^{3} \left(40 x^{3} \cot{\left(x \right)} - \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{\left(2 x^{4} + 5\right) \cot{\left(x \right)}} + \frac{240 x^{3} \left(640 x^{6} \cot{\left(x \right)} - 40 x^{3} \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right) + 60 x^{2} \left(2 x^{4} + 5\right) \cot{\left(x \right)} + \left(2 x^{4} + 5\right)^{2} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right)}{\left(2 x^{4} + 5\right)^{2}} + \frac{120 x^{2} \left(40 x^{3} \cot{\left(x \right)} - \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right)}{2 x^{4} + 5} + \frac{\left(\frac{40 x^{3}}{2 x^{4} + 5} - \frac{\cot^{2}{\left(x \right)} + 1}{\cot{\left(x \right)}}\right) \left(40 x^{3} \cot{\left(x \right)} - \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot{\left(x \right)}} - \frac{3 \left(40 x^{3} \cot{\left(x \right)} - \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right) \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{2}{\left(x \right)}} + 2 \left(40 x^{3} \cot{\left(x \right)} - \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right) \left(\cot^{2}{\left(x \right)} + 1\right) + 2 \left(40 x^{3} \cot{\left(x \right)} - \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right) \left(- \frac{960 x^{6}}{\left(2 x^{4} + 5\right)^{2}} + \frac{40 x^{3} \left(\cot^{2}{\left(x \right)} + 1\right)}{\left(2 x^{4} + 5\right) \cot{\left(x \right)}} + \frac{60 x^{2}}{2 x^{4} + 5} - \frac{\left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{2}{\left(x \right)}} + \cot^{2}{\left(x \right)} + 1\right) + \frac{2 \left(\frac{40 x^{3}}{2 x^{4} + 5} - \frac{\cot^{2}{\left(x \right)} + 1}{\cot{\left(x \right)}}\right) \left(640 x^{6} \cot{\left(x \right)} - 40 x^{3} \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right) + 60 x^{2} \left(2 x^{4} + 5\right) \cot{\left(x \right)} + \left(2 x^{4} + 5\right)^{2} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right)}{2 x^{4} + 5} - \frac{6 \left(\cot^{2}{\left(x \right)} + 1\right) \left(640 x^{6} \cot{\left(x \right)} - 40 x^{3} \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right) + 60 x^{2} \left(2 x^{4} + 5\right) \cot{\left(x \right)} + \left(2 x^{4} + 5\right)^{2} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right)}{\left(2 x^{4} + 5\right) \cot{\left(x \right)}} - \frac{2 \left(15360 x^{9} \cot{\left(x \right)} - 1920 x^{6} \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right) + 5760 x^{5} \left(2 x^{4} + 5\right) \cot{\left(x \right)} + 120 x^{3} \left(2 x^{4} + 5\right)^{2} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - 180 x^{2} \left(2 x^{4} + 5\right)^{2} \left(\cot^{2}{\left(x \right)} + 1\right) + 120 x \left(2 x^{4} + 5\right)^{2} \cot{\left(x \right)} - \left(2 x^{4} + 5\right)^{3} \left(\cot^{2}{\left(x \right)} + 1\right)^{2} - 2 \left(2 x^{4} + 5\right)^{3} \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)}\right)}{\left(2 x^{4} + 5\right)^{2}}\right) \log{\left(x \right)}^{3}}{\left(2 x^{4} + 5\right) \cot{\left(x \right)}} - \frac{3 \left(40 x^{3} \cot{\left(x \right)} - \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right) \left(- \frac{5 x^{4} \left(\frac{5 x^{5}}{x^{5} + 9} - 8\right) \log{\left(x \right)}^{2}}{\sqrt{x^{5} + 9}} + \frac{20 x^{4} \left(\log{\left(x \right)} + 3\right) \log{\left(x \right)}}{\sqrt{x^{5} + 9}} + \frac{12 \sqrt{x^{5} + 9} \left(\log{\left(x \right)} + 2\right)}{x}\right) \log{\left(x \right)}}{4 \left(2 x^{4} + 5\right) \cot{\left(x \right)}} + \frac{3 \left(\frac{5 x^{5} \log{\left(x \right)}}{\sqrt{x^{5} + 9}} + 2 \sqrt{x^{5} + 9} \left(\log{\left(x \right)} + 3\right)\right) \left(\frac{40 x^{3} \left(40 x^{3} \cot{\left(x \right)} - \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right)}{2 x^{4} + 5} + \left(\frac{40 x^{3}}{2 x^{4} + 5} - \frac{\cot^{2}{\left(x \right)} + 1}{\cot{\left(x \right)}}\right) \left(40 x^{3} \cot{\left(x \right)} - \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right) - \frac{\left(40 x^{3} \cot{\left(x \right)} - \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right)\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot{\left(x \right)}} - \frac{2 \left(640 x^{6} \cot{\left(x \right)} - 40 x^{3} \left(2 x^{4} + 5\right) \left(\cot^{2}{\left(x \right)} + 1\right) + 60 x^{2} \left(2 x^{4} + 5\right) \cot{\left(x \right)} + \left(2 x^{4} + 5\right)^{2} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right)}{2 x^{4} + 5}\right) \log{\left(x \right)}^{2}}{2 \left(2 x^{4} + 5\right) \cot{\left(x \right)}} - \frac{3 \sqrt{x^{5} + 9} \left(3 \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)} - 2 \log{\left(x \right)}^{2} + 6 \log{\left(x \right)} - 2\right)}{x^{2}}}{\left(2 x^{4} + 5\right)^{5} \cot{\left(x \right)}}$$

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Gráfico

Derivada de (xln^3(x)sqrt(x^5+9))/(ctgx(2x^4+5)^5)

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Derivada de (xln^3(x)sqrt(x^5+9))/(ctgx(2x^4+5)^5)

Gráfico:

Definida a trozos:

Solución

Method #1

Method #2