Derivada de x^xlogx(sinx)^sinx

Solución

Ha introducido [src]

 x           sin(x)   
x *log(x)*sin      (x)

$$x^{x} \log{\left(x \right)} \sin^{\sin{\left(x \right)}}{\left(x \right)}$$

(x^x*log(x))*sin(x)^sin(x)

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = x^{x} \log{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{x}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. No logro encontrar los pasos en la búsqueda de esta derivada.
    
    Perola derivada
    
    $x^{x} \left(\log{\left(x \right)} + 1\right)$
  $g{\left(x \right)} = \log{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Derivado $\log{\left(x \right)}$ es $\frac{1}{x}$ .
  Como resultado de: $x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x}}{x}$
$g{\left(x \right)} = \sin^{\sin{\left(x \right)}}{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin^{\sin{\left(x \right)}}{\left(x \right)}$
Como resultado de: $x^{x} \left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \log{\left(x \right)} \sin^{\sin{\left(x \right)}}{\left(x \right)} + \left(x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x}}{x}\right) \sin^{\sin{\left(x \right)}}{\left(x \right)}$
Simplificamos:

$x^{x - 1} \left(x \log{\left(x \right)}^{2} + x \log{\left(x \right)} \log{\left(\sin{\left(x \right)} \right)} + 2 x \log{\left(x \right)} + 1\right) \sin^{\sin{\left(x \right)}}{\left(x \right)}$

Respuesta:

$x^{x - 1} \left(x \log{\left(x \right)}^{2} + x \log{\left(x \right)} \log{\left(\sin{\left(x \right)} \right)} + 2 x \log{\left(x \right)} + 1\right) \sin^{\sin{\left(x \right)}}{\left(x \right)}$

Gráfica

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Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

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Derivada de x^xlogx(sinx)^sinx

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