Derivada de (x*sin(x*loge(x)))^1/2

Ha introducido [src]

     _________________
    /      /x*log(x)\ 
   /  x*sin|--------| 
  /        |   / 1\ | 
\/         \log\e / /

\sqrt{x \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}

sqrt(x*sin((x*log(x))/log(exp(1))))

Solución detallada

Sustituimos $u = x \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}$ .
Según el principio, aplicamos: $\sqrt{u}$ tenemos $\frac{1}{2 \sqrt{u}}$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} x \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Según el principio, aplicamos: $x$ tenemos $1$
  $g{\left(x \right)} = \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Sustituimos $u = \frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}}$ .
  2. La derivada del seno es igual al coseno:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}}$ :
    1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
      1. Se aplica la regla de la derivada de una multiplicación:
        
        $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
        
        $f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
        
        Según el principio, aplicamos: $x$ tenemos $1$
        
        $g{\left(x \right)} = \log{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
        
        Derivado $\log{\left(x \right)}$ es $\frac{1}{x}$ .
        
        Como resultado de: $\log{\left(x \right)} + 1$
      Entonces, como resultado: $\frac{\log{\left(x \right)} + 1}{\log{\left(e^{1} \right)}}$
    Como resultado de la secuencia de reglas:
    
    $\frac{\left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}}$
  Como resultado de: $\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}$
Como resultado de la secuencia de reglas:

$\frac{\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{2 \sqrt{x \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}}$
Simplificamos:

$\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(x \log{\left(x \right)} \right)} + \sin{\left(x \log{\left(x \right)} \right)}}{2 \sqrt{x \sin{\left(x \log{\left(x \right)} \right)}}}$

Respuesta:

$\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(x \log{\left(x \right)} \right)} + \sin{\left(x \log{\left(x \right)} \right)}}{2 \sqrt{x \sin{\left(x \log{\left(x \right)} \right)}}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

                       /   /x*log(x)\                     /x*log(x)\\
                       |sin|--------|   x*(1 + log(x))*cos|--------||
     _________________ |   |   / 1\ |                     |   / 1\ ||
    /      /x*log(x)\  |   \log\e / /                     \log\e / /|
   /  x*sin|--------| *|------------- + ----------------------------|
  /        |   / 1\ |  |      2                       / 1\          |
\/         \log\e / /  \                         2*log\e /          /
---------------------------------------------------------------------
                                /x*log(x)\                           
                           x*sin|--------|                           
                                |   / 1\ |                           
                                \log\e / /

\frac{\sqrt{x \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}} \left(\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{2 \log{\left(e^{1} \right)}} + \frac{\sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{2}\right)}{x \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}

Simplificar

Segunda derivada [src]

                       /                                                                                                                                                                                      2                                                                              \
                       |    /                  /x*log(x)\                \     /                                             2    /x*log(x)\                \   /                  /x*log(x)\                \                   /                  /x*log(x)\                \              |
                       |    |x*(1 + log(x))*cos|--------|                |     |                               x*(1 + log(x)) *sin|--------|                |   |x*(1 + log(x))*cos|--------|                |                   |x*(1 + log(x))*cos|--------|                |              |
                       |    |                  |   / 1\ |                |     |                                                  |   / 1\ |                |   |                  |   / 1\ |                |                   |                  |   / 1\ |                |              |
                       |    |                  \log\e / /      /x*log(x)\|     |                  /x*log(x)\                      \log\e / /      /x*log(x)\|   |                  \log\e / /      /x*log(x)\|                   |                  \log\e / /      /x*log(x)\|    /x*log(x)\|
                       |  2*|---------------------------- + sin|--------||   2*|2*(1 + log(x))*cos|--------| - ----------------------------- + cos|--------||   |---------------------------- + sin|--------||    2*(1 + log(x))*|---------------------------- + sin|--------||*cos|--------||
     _________________ |    |             / 1\                 |   / 1\ ||     |                  |   / 1\ |                 / 1\                 |   / 1\ ||   |             / 1\                 |   / 1\ ||                   |             / 1\                 |   / 1\ ||    |   / 1\ ||
    /      /x*log(x)\  |    \          log\e /                 \log\e / //     \                  \log\e / /              log\e /                 \log\e / //   \          log\e /                 \log\e / //                   \          log\e /                 \log\e / //    \log\e / /|
   /  x*sin|--------| *|- ------------------------------------------------ + -------------------------------------------------------------------------------- + ----------------------------------------------- - ---------------------------------------------------------------------------|
  /        |   / 1\ |  |                         x                                                                  / 1\                                                             /x*log(x)\                                                 / 1\    /x*log(x)\                           |
\/         \log\e / /  |                                                                                         log\e /                                                        x*sin|--------|                                              log\e /*sin|--------|                           |
                       |                                                                                                                                                             |   / 1\ |                                                         |   / 1\ |                           |
                       \                                                                                                                                                             \log\e / /                                                         \log\e / /                           /
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                             /x*log(x)\                                                                                                                                       
                                                                                                                                      4*x*sin|--------|                                                                                                                                       
                                                                                                                                             |   / 1\ |                                                                                                                                       
                                                                                                                                             \log\e / /

\frac{\sqrt{x \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}} \left(- \frac{2 \left(\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}\right) \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)} \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}} + \frac{2 \left(- \frac{x \left(\log{\left(x \right)} + 1\right)^{2} \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + 2 \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)} + \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}\right)}{\log{\left(e^{1} \right)}} + \frac{\left(\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}\right)^{2}}{x \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}} - \frac{2 \left(\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}\right)}{x}\right)}{4 x \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}

Simplificar

Tercera derivada [src]

                       /                                                                                                                                                                                                                                                                                                                                                                2                                                 3                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          2                           \
                       |                  /x*log(x)\                          /x*log(x)\                 2    /x*log(x)\                     /x*log(x)\                 3    /x*log(x)\                 /                  /x*log(x)\                \                                                2    /x*log(x)\                     /                  /x*log(x)\                \    /                  /x*log(x)\                \                                 /                  /x*log(x)\                \                /                                             2    /x*log(x)\                \                 /                  /x*log(x)\                \                   /                  /x*log(x)\                \ /                                             2    /x*log(x)\                \                /                  /x*log(x)\                \                   /                  /x*log(x)\                \                            |
                       |x*(1 + log(x))*cos|--------|                     2*cos|--------|   3*(1 + log(x)) *sin|--------|   3*(1 + log(x))*sin|--------|   x*(1 + log(x)) *cos|--------|                 |x*(1 + log(x))*cos|--------|                |                                  x*(1 + log(x)) *sin|--------|                     |x*(1 + log(x))*cos|--------|                |    |x*(1 + log(x))*cos|--------|                |                                 |x*(1 + log(x))*cos|--------|                |                |                               x*(1 + log(x)) *sin|--------|                |                 |x*(1 + log(x))*cos|--------|                |                   |x*(1 + log(x))*cos|--------|                | |                               x*(1 + log(x)) *sin|--------|                |                |x*(1 + log(x))*cos|--------|                |                   |x*(1 + log(x))*cos|--------|                |                            |
                       |                  |   / 1\ |                          |   / 1\ |                      |   / 1\ |                     |   / 1\ |                      |   / 1\ |                 |                  |   / 1\ |                |                                                     |   / 1\ |                     |                  |   / 1\ |                |    |                  |   / 1\ |                |                                 |                  |   / 1\ |                |                |                                                  |   / 1\ |                |                 |                  |   / 1\ |                |                   |                  |   / 1\ |                | |                                                  |   / 1\ |                |                |                  |   / 1\ |                |                   |                  |   / 1\ |                |                            |
                       |                  \log\e / /      /x*log(x)\          \log\e / /                      \log\e / /                     \log\e / /                      \log\e / /               2 |                  \log\e / /      /x*log(x)\|                     /x*log(x)\                      \log\e / /      /x*log(x)\     |                  \log\e / /      /x*log(x)\|    |                  \log\e / /      /x*log(x)\|                2    2/x*log(x)\ |                  \log\e / /      /x*log(x)\|                |                  /x*log(x)\                      \log\e / /      /x*log(x)\|    /x*log(x)\   |                  \log\e / /      /x*log(x)\|    /x*log(x)\     |                  \log\e / /      /x*log(x)\| |                  /x*log(x)\                      \log\e / /      /x*log(x)\|                |                  \log\e / /      /x*log(x)\|    /x*log(x)\     |                  \log\e / /      /x*log(x)\|                  /x*log(x)\|
                       |---------------------------- + sin|--------|   - --------------- + ----------------------------- + ---------------------------- + -----------------------------   (1 + log(x)) *|---------------------------- + sin|--------||   2*(1 + log(x))*cos|--------| - ----------------------------- + cos|--------|   3*|---------------------------- + sin|--------||    |---------------------------- + sin|--------||    (1 + log(x)) *cos |--------|*|---------------------------- + sin|--------||   (1 + log(x))*|2*(1 + log(x))*cos|--------| - ----------------------------- + cos|--------||*cos|--------|   |---------------------------- + sin|--------||*cos|--------|   3*|---------------------------- + sin|--------||*|2*(1 + log(x))*cos|--------| - ----------------------------- + cos|--------||   (1 + log(x))*|---------------------------- + sin|--------||*cos|--------|   3*|---------------------------- + sin|--------|| *(1 + log(x))*cos|--------||
     _________________ |             / 1\                 |   / 1\ |            x                        / 1\                           / 1\                            2/ 1\                           |             / 1\                 |   / 1\ ||                     |   / 1\ |                 / 1\                 |   / 1\ |     |             / 1\                 |   / 1\ ||    |             / 1\                 |   / 1\ ||                      |   / 1\ | |             / 1\                 |   / 1\ ||                |                  |   / 1\ |                 / 1\                 |   / 1\ ||    |   / 1\ |   |             / 1\                 |   / 1\ ||    |   / 1\ |     |             / 1\                 |   / 1\ || |                  |   / 1\ |                 / 1\                 |   / 1\ ||                |             / 1\                 |   / 1\ ||    |   / 1\ |     |             / 1\                 |   / 1\ ||                  |   / 1\ ||
    /      /x*log(x)\  |          log\e /                 \log\e / /                                  log\e /                        log\e /                         log \e /                           \          log\e /                 \log\e / //                     \log\e / /              log\e /                 \log\e / /     \          log\e /                 \log\e / //    \          log\e /                 \log\e / //                      \log\e / / \          log\e /                 \log\e / //                \                  \log\e / /              log\e /                 \log\e / //    \log\e / /   \          log\e /                 \log\e / //    \log\e / /     \          log\e /                 \log\e / // \                  \log\e / /              log\e /                 \log\e / //                \          log\e /                 \log\e / //    \log\e / /     \          log\e /                 \log\e / //                  \log\e / /|
   /  x*sin|--------| *|-------------------------------------------- - ---------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------ - ---------------------------------------------------------------------------- - ------------------------------------------------- + ----------------------------------------------- + --------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------- - ----------------------------------------------------------------------------|
  /        |   / 1\ |  |                      2                                                                                / 1\                                                                                     2/ 1\                                                                  / 1\                                                        2    /x*log(x)\                                   2    2/x*log(x)\                                              2/ 1\    2/x*log(x)\                                                                          2/ 1\    /x*log(x)\                                                                    / 1\    /x*log(x)\                                                                               / 1\    /x*log(x)\                                                                                    / 1\    /x*log(x)\                                                            / 1\    2/x*log(x)\                         |
\/         \log\e / /  |                     x                                                                            2*log\e /                                                                                2*log \e /                                                             x*log\e /                                                     4*x *sin|--------|                                8*x *sin |--------|                                           log \e /*sin |--------|                                                                       log \e /*sin|--------|                                                             2*x*log\e /*sin|--------|                                                                        4*x*log\e /*sin|--------|                                                                               x*log\e /*sin|--------|                                                     4*x*log\e /*sin |--------|                         |
                       |                                                                                                                                                                                                                                                                                                                                        |   / 1\ |                                         |   / 1\ |                                                        |   / 1\ |                                                                                   |   / 1\ |                                                                            |   / 1\ |                                                                                       |   / 1\ |                                                                                            |   / 1\ |                                                                     |   / 1\ |                         |
                       \                                                                                                                                                                                                                                                                                                                                        \log\e / /                                         \log\e / /                                                        \log\e / /                                                                                   \log\e / /                                                                            \log\e / /                                                                                       \log\e / /                                                                                            \log\e / /                                                                     \log\e / /                         /
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              /x*log(x)\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         x*sin|--------|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |   / 1\ |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              \log\e / /

\frac{\sqrt{x \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}} \left(\frac{\left(\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}\right) \left(\log{\left(x \right)} + 1\right)^{2}}{2 \log{\left(e^{1} \right)}^{2}} + \frac{\left(\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}\right) \left(\log{\left(x \right)} + 1\right)^{2} \cos^{2}{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}^{2} \sin^{2}{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}} - \frac{\left(\log{\left(x \right)} + 1\right) \left(- \frac{x \left(\log{\left(x \right)} + 1\right)^{2} \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + 2 \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)} + \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}^{2} \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}} - \frac{\frac{x \left(\log{\left(x \right)} + 1\right)^{3} \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}^{2}} + \frac{3 \left(\log{\left(x \right)} + 1\right)^{2} \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + \frac{3 \left(\log{\left(x \right)} + 1\right) \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} - \frac{2 \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{x}}{2 \log{\left(e^{1} \right)}} - \frac{3 \left(\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}\right)^{2} \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{4 x \log{\left(e^{1} \right)} \sin^{2}{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}} + \frac{\left(\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}\right) \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{x \log{\left(e^{1} \right)} \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}} + \frac{3 \left(\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}\right) \left(- \frac{x \left(\log{\left(x \right)} + 1\right)^{2} \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + 2 \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)} + \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}\right)}{4 x \log{\left(e^{1} \right)} \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}} - \frac{\left(\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{2 x \log{\left(e^{1} \right)} \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}} - \frac{- \frac{x \left(\log{\left(x \right)} + 1\right)^{2} \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + 2 \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)} + \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{x \log{\left(e^{1} \right)}} + \frac{\left(\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}\right)^{3}}{8 x^{2} \sin^{2}{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}} - \frac{3 \left(\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}\right)^{2}}{4 x^{2} \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}} + \frac{\frac{x \left(\log{\left(x \right)} + 1\right) \cos{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{\log{\left(e^{1} \right)}} + \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}{x^{2}}\right)}{x \sin{\left(\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}} \right)}}

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Derivada de (xsin(xloge(x)))^1/2

Gráfico:

Definida a trozos:

Solución