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cos(asin(x))
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Derivada de la función/ x^(sin(x)-1)*sin(x)+x^(sin(x))*ln(x)*cos(x)

Derivada de x^(sin(x)-1)sin(x)+x^(sin(x))ln(x)*cos(x)

Solución

Ha introducido [src]

 sin(x) - 1           sin(x)              
x          *sin(x) + x      *log(x)*cos(x)

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

x^(sin(x) - 1)*sin(x) + (x^sin(x)*log(x))*cos(x)

Solución detallada

diferenciamos $x^{\sin{\left(x \right)} - 1} \sin{\left(x \right)} + x^{\sin{\left(x \right)}} \log{\left(x \right)} \cos{\left(x \right)}$ miembro por miembro:
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{\sin{\left(x \right)} - 1}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. No logro encontrar los pasos en la búsqueda de esta derivada.
    
    Perola derivada
    
    $\left(\log{\left(\sin{\left(x \right)} - 1 \right)} + 1\right) \left(\sin{\left(x \right)} - 1\right)^{\sin{\left(x \right)} - 1}$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. La derivada del seno es igual al coseno:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  Como resultado de: $x^{\sin{\left(x \right)} - 1} \cos{\left(x \right)} + \left(\log{\left(\sin{\left(x \right)} - 1 \right)} + 1\right) \left(\sin{\left(x \right)} - 1\right)^{\sin{\left(x \right)} - 1} \sin{\left(x \right)}$
2. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{\sin{\left(x \right)}} \log{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Se aplica la regla de la derivada de una multiplicación:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x^{\sin{\left(x \right)}}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
    1. No logro encontrar los pasos en la búsqueda de esta derivada.
      
      Perola derivada
      
      $\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin^{\sin{\left(x \right)}}{\left(x \right)}$
    $g{\left(x \right)} = \log{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
    1. Derivado $\log{\left(x \right)}$ es $\frac{1}{x}$ .
    Como resultado de: $\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \log{\left(x \right)} \sin^{\sin{\left(x \right)}}{\left(x \right)} + \frac{x^{\sin{\left(x \right)}}}{x}$
  $g{\left(x \right)} = \cos{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. La derivada del coseno es igual a menos el seno:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  Como resultado de: $- x^{\sin{\left(x \right)}} \log{\left(x \right)} \sin{\left(x \right)} + \left(\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \log{\left(x \right)} \sin^{\sin{\left(x \right)}}{\left(x \right)} + \frac{x^{\sin{\left(x \right)}}}{x}\right) \cos{\left(x \right)}$
Como resultado de: $x^{\sin{\left(x \right)} - 1} \cos{\left(x \right)} - x^{\sin{\left(x \right)}} \log{\left(x \right)} \sin{\left(x \right)} + \left(\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \log{\left(x \right)} \sin^{\sin{\left(x \right)}}{\left(x \right)} + \frac{x^{\sin{\left(x \right)}}}{x}\right) \cos{\left(x \right)} + \left(\log{\left(\sin{\left(x \right)} - 1 \right)} + 1\right) \left(\sin{\left(x \right)} - 1\right)^{\sin{\left(x \right)} - 1} \sin{\left(x \right)}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

Gráfico

Derivada de x^(sin(x)-1)*sin(x)+x^(sin(x))*ln(x)*cos(x)

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

Derivada de x^(sin(x)-1)sin(x)+x^(sin(x))ln(x)*cos(x)

Gráfico:

Definida a trozos:

Solución

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

Derivada de x^(sin(x)-1)*sin(x)+x^(sin(x))*ln(x)*cos(x)

Gráfico:

Definida a trozos:

Solución

Derivada de x^(sin(x)-1)sin(x)+x^(sin(x))ln(x)*cos(x)