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Derivada de la función/ x((x*cos(x))^(x*cos(x))-(x^2)*cos(x)*sin(x))

Derivada de x((xcos(x))^(xcos(x))-(x^2)cos(x)sin(x))

Solución

Ha introducido [src]

  /          x*cos(x)    2              \
x*\(x*cos(x))         - x *cos(x)*sin(x)/

$$x \left(\left(x \cos{\left(x \right)}\right)^{x \cos{\left(x \right)}} - x^{2} \cos{\left(x \right)} \sin{\left(x \right)}\right)$$

x*((x*cos(x))^(x*cos(x)) - x^2*cos(x)*sin(x))

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Según el principio, aplicamos: $x$ tenemos $1$
$g{\left(x \right)} = \left(x \cos{\left(x \right)}\right)^{x \cos{\left(x \right)}} - x^{2} \cos{\left(x \right)} \sin{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. diferenciamos $\left(x \cos{\left(x \right)}\right)^{x \cos{\left(x \right)}} - x^{2} \cos{\left(x \right)} \sin{\left(x \right)}$ miembro por miembro:
  1. No logro encontrar los pasos en la búsqueda de esta derivada.
    
    Perola derivada
    
    $\left(x \cos{\left(x \right)}\right)^{x \cos{\left(x \right)}} \left(\log{\left(x \cos{\left(x \right)} \right)} + 1\right)$
  2. 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:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} h{\left(x \right)} = f{\left(x \right)} g{\left(x \right)} \frac{d}{d x} h{\left(x \right)} + f{\left(x \right)} h{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} h{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = x^{2}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
      1. Según el principio, aplicamos: $x^{2}$ tenemos $2 x$
      $g{\left(x \right)} = \cos{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
      1. La derivada del coseno es igual a menos el seno:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      $h{\left(x \right)} = \sin{\left(x \right)}$ ; calculamos $\frac{d}{d x} h{\left(x \right)}$ :
      1. La derivada del seno es igual al coseno:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      Como resultado de: $- x^{2} \sin^{2}{\left(x \right)} + x^{2} \cos^{2}{\left(x \right)} + 2 x \sin{\left(x \right)} \cos{\left(x \right)}$
    Entonces, como resultado: $x^{2} \sin^{2}{\left(x \right)} - x^{2} \cos^{2}{\left(x \right)} - 2 x \sin{\left(x \right)} \cos{\left(x \right)}$
  Como resultado de: $x^{2} \sin^{2}{\left(x \right)} - x^{2} \cos^{2}{\left(x \right)} - 2 x \sin{\left(x \right)} \cos{\left(x \right)} + \left(x \cos{\left(x \right)}\right)^{x \cos{\left(x \right)}} \left(\log{\left(x \cos{\left(x \right)} \right)} + 1\right)$
Como resultado de: $x \left(x^{2} \sin^{2}{\left(x \right)} - x^{2} \cos^{2}{\left(x \right)} - 2 x \sin{\left(x \right)} \cos{\left(x \right)} + \left(x \cos{\left(x \right)}\right)^{x \cos{\left(x \right)}} \left(\log{\left(x \cos{\left(x \right)} \right)} + 1\right)\right) + \left(x \cos{\left(x \right)}\right)^{x \cos{\left(x \right)}} - x^{2} \cos{\left(x \right)} \sin{\left(x \right)}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

  /                                 /                                                                                  2                       2                                                      \                                                                                                                                                                                                                                                                                                                               /                                                                                2\                    \                                      /                                                                                2\                                                                                                                                                                                           
  |       2                x*cos(x) |                                                              (-cos(x) + x*sin(x))    (-cos(x) + x*sin(x)) *sin(x)   3*(-cos(x) + x*sin(x))*(2*sin(x) + x*cos(x))|   /            2                    \                    x*cos(x)                                                          3      2    2         2    2        /             2                    \                                                        x*cos(x)                                                           |                                                            (-cos(x) + x*sin(x)) |                    |          2                  x*cos(x) |                                                            (-cos(x) + x*sin(x)) |               x*cos(x)                                                          2     /             2                    \                                                 2              
x*|- 2*cos (x) + (x*cos(x))        *|-3*cos(x) + x*sin(x) + (-3*cos(x) + x*sin(x))*log(x*cos(x)) - --------------------- + ---------------------------- + --------------------------------------------| + \6*sin(x) - x *sin(x) + 6*x*cos(x)/*sin(x) - (x*cos(x))        *(-cos(x) + x*sin(x) + (-cos(x) + x*sin(x))*log(x*cos(x)))  - 2*x *sin (x) + 2*x *cos (x) + 2*\-2*cos(x) + x *cos(x) + 4*x*sin(x)/*cos(x) - x*(-2*cos(x) + x*sin(x))*sin(x) + 3*(x*cos(x))        *(-cos(x) + x*sin(x) + (-cos(x) + x*sin(x))*log(x*cos(x)))*|2*sin(x) + x*cos(x) + (2*sin(x) + x*cos(x))*log(x*cos(x)) - ---------------------| + 8*x*cos(x)*sin(x)| - 6*x*cos (x) - 3*(x*cos(x))        *|2*sin(x) + x*cos(x) + (2*sin(x) + x*cos(x))*log(x*cos(x)) - ---------------------| + 3*(x*cos(x))        *(-cos(x) + x*sin(x) + (-cos(x) + x*sin(x))*log(x*cos(x)))  + 3*\-2*cos(x) + x *cos(x) + 4*x*sin(x)/*sin(x) + 3*x*(-2*cos(x) + x*sin(x))*cos(x) + 6*x *cos(x)*sin(x)
  |                                 |                                                                     2                              2                                  x*cos(x)                  |                                                                                                                                                                                                                                                                                                                               \                                                                   x*cos(x)      /                    |                                      \                                                                   x*cos(x)      /                                                                                                                                                                                           
  \                                 \                                                                    x *cos(x)                  x*cos (x)                                                         /                                                                                                                                                                                                                                                                                                                                                                                                                                      /

$$6 x^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 3 x \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \cos{\left(x \right)} + x \left(- 2 x^{2} \sin^{2}{\left(x \right)} + 2 x^{2} \cos^{2}{\left(x \right)} - x \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin{\left(x \right)} + 8 x \sin{\left(x \right)} \cos{\left(x \right)} - \left(x \cos{\left(x \right)}\right)^{x \cos{\left(x \right)}} \left(x \sin{\left(x \right)} + \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(x \cos{\left(x \right)} \right)} - \cos{\left(x \right)}\right)^{3} + 3 \left(x \cos{\left(x \right)}\right)^{x \cos{\left(x \right)}} \left(x \sin{\left(x \right)} + \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(x \cos{\left(x \right)} \right)} - \cos{\left(x \right)}\right) \left(x \cos{\left(x \right)} + \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \log{\left(x \cos{\left(x \right)} \right)} + 2 \sin{\left(x \right)} - \frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}{x \cos{\left(x \right)}}\right) + \left(x \cos{\left(x \right)}\right)^{x \cos{\left(x \right)}} \left(x \sin{\left(x \right)} + \left(x \sin{\left(x \right)} - 3 \cos{\left(x \right)}\right) \log{\left(x \cos{\left(x \right)} \right)} - 3 \cos{\left(x \right)} + \frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2} \sin{\left(x \right)}}{x \cos^{2}{\left(x \right)}} + \frac{3 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x \cos{\left(x \right)}} - \frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}{x^{2} \cos{\left(x \right)}}\right) + \left(- x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} + 6 \sin{\left(x \right)}\right) \sin{\left(x \right)} + 2 \left(x^{2} \cos{\left(x \right)} + 4 x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \cos{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) - 6 x \cos^{2}{\left(x \right)} + 3 \left(x \cos{\left(x \right)}\right)^{x \cos{\left(x \right)}} \left(x \sin{\left(x \right)} + \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(x \cos{\left(x \right)} \right)} - \cos{\left(x \right)}\right)^{2} - 3 \left(x \cos{\left(x \right)}\right)^{x \cos{\left(x \right)}} \left(x \cos{\left(x \right)} + \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \log{\left(x \cos{\left(x \right)} \right)} + 2 \sin{\left(x \right)} - \frac{\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}{x \cos{\left(x \right)}}\right) + 3 \left(x^{2} \cos{\left(x \right)} + 4 x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin{\left(x \right)}$$

Simplificar

Gráfico

Derivada de x((x*cos(x))^(x*cos(x))-(x^2)*cos(x)*sin(x))

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

Derivada de x((xcos(x))^(xcos(x))-(x^2)cos(x)sin(x))

Gráfico:

Definida a trozos:

Solución

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

Derivada de x((x*cos(x))^(x*cos(x))-(x^2)*cos(x)*sin(x))

Gráfico:

Definida a trozos:

Solución

Derivada de x((xcos(x))^(xcos(x))-(x^2)cos(x)sin(x))