Derivada de -((xsin(x)+cos(x))cos(cosx/x))/x^2

Ha introducido [src]

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

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

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

Solución detallada

Se aplica la regla de la derivada parcial:

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

$f{\left(x \right)} = - \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \cos{\left(\frac{\cos{\left(x \right)}}{x} \right)}$ y $g{\left(x \right)} = x^{2}$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
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:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x \sin{\left(x \right)} + \cos{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
    1. diferenciamos $x \sin{\left(x \right)} + \cos{\left(x \right)}$ miembro por miembro:
      1. Se aplica la regla de la derivada de una multiplicación:
        
        $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
        
        $f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
        
        Según el principio, aplicamos: $x$ tenemos $1$
        
        $g{\left(x \right)} = \sin{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
        
        La derivada del seno es igual al coseno:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
        
        Como resultado de: $x \cos{\left(x \right)} + \sin{\left(x \right)}$
      2. 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: $x \cos{\left(x \right)}$
    $g{\left(x \right)} = \cos{\left(\frac{\cos{\left(x \right)}}{x} \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
    1. Sustituimos $u = \frac{\cos{\left(x \right)}}{x}$ .
    2. La derivada del coseno es igual a menos el seno:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
    3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{\cos{\left(x \right)}}{x}$ :
      1. 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 $g{\left(x \right)} = x$ .
        
        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 $\frac{d}{d x} g{\left(x \right)}$ :
        
        Según el principio, aplicamos: $x$ tenemos $1$
        
        Ahora aplicamos la regla de la derivada de una divesión:
        
        $\frac{- x \sin{\left(x \right)} - \cos{\left(x \right)}}{x^{2}}$
      Como resultado de la secuencia de reglas:
      
      $- \frac{\left(- x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(\frac{\cos{\left(x \right)}}{x} \right)}}{x^{2}}$
    Como resultado de: $x \cos{\left(x \right)} \cos{\left(\frac{\cos{\left(x \right)}}{x} \right)} - \frac{\left(- x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(\frac{\cos{\left(x \right)}}{x} \right)}}{x^{2}}$
  Entonces, como resultado: $- x \cos{\left(x \right)} \cos{\left(\frac{\cos{\left(x \right)}}{x} \right)} + \frac{\left(- x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(\frac{\cos{\left(x \right)}}{x} \right)}}{x^{2}}$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. Según el principio, aplicamos: $x^{2}$ tenemos $2 x$
Ahora aplicamos la regla de la derivada de una divesión:

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

                                                                                                                      /                                                       /                                                               2            \                                         \                                                                                                                                                                                                                                                                                                                                                                         
                                                                                                                      |                                                       |                                              /cos(x)         \     /cos(x)\|                                         |                                                                                                                                                                                                                                                                                                                                                                         
                                                                                                                      |                                                       |                                              |------ + sin(x)| *cos|------||                                         |                                                                                                                 /                                                                           3                 /cos(x)         \ /          2*sin(x)   2*cos(x)\    /cos(x)\\                                                                                            
                                                                                                                      |                                                       |/          2*sin(x)   2*cos(x)\    /cos(x)\   \  x            /     \  x   /|                                         |                                                                                                                 |                                                          /cos(x)         \     /cos(x)\   3*|------ + sin(x)|*|-cos(x) + -------- + --------|*cos|------||                                                                                            
                                       /                                           /cos(x)         \    /cos(x)\\     |                                   (x*sin(x) + cos(x))*||-cos(x) + -------- + --------|*sin|------| + ------------------------------|                                         |                                                                                                                 |                                                          |------ + sin(x)| *sin|------|     \  x            / |             x           2   |    \  x   /|                                                                                            
                                       |                       (x*sin(x) + cos(x))*|------ + sin(x)|*sin|------||     |                                                       ||             x           2   |    \  x   /                 x               |                                         |     /                                                               2            \                              |/  6*cos(x)   6*sin(x)   3*cos(x)         \    /cos(x)\   \  x            /     \  x   /                       \                        x    /            |                                                                                            
                                       |            /cos(x)\                       \  x            /    \  x   /|     |                        /cos(x)\                       \\                        x    /                                             /     /cos(x)         \           /cos(x)\|     |                                              /cos(x)         \     /cos(x)\|          (x*sin(x) + cos(x))*||- -------- - -------- + -------- + sin(x)|*sin|------| + ------------------------------ - ---------------------------------------------------------------|                             /cos(x)\                          /cos(x)         \    /cos(x)\
                                    18*|x*cos(x)*cos|------| + -------------------------------------------------|   6*|(-cos(x) + x*sin(x))*cos|------| + -------------------------------------------------------------------------------------------------- - 2*|------ + sin(x)|*cos(x)*sin|------||     |                                              |------ + sin(x)| *cos|------||                              ||      3          2         x             |    \  x   /                  2                                                x                               |   24*(x*sin(x) + cos(x))*cos|------|   3*(-cos(x) + x*sin(x))*|------ + sin(x)|*sin|------|
                         /cos(x)\      \            \  x   /                           x                        /     \                        \  x   /                                                   x                                                      \  x            /           \  x   //     |/          2*sin(x)   2*cos(x)\    /cos(x)\   \  x            /     \  x   /|                              \\     x          x                        /                             x                                                                                 /                             \  x   /                          \  x            /    \  x   /
(2*sin(x) + x*cos(x))*cos|------| - ----------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + 3*||-cos(x) + -------- + --------|*sin|------| + ------------------------------|*cos(x) + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------- + ----------------------------------------------------
                         \  x   /                                          2                                                                                                                                x                                                                                              ||             x           2   |    \  x   /                 x               |                                                                                                 x                                                                                                            3                                            x                          
                                                                          x                                                                                                                                                                                                                                \\                        x    /                                             /                                                                                                                                                                                                             x                                                                        
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                        2                                                                                                                                                                                                                                                                                                                                      
                                                                                                                                                                                                                                                                                                                                       x

\frac{\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos{\left(\frac{\cos{\left(x \right)}}{x} \right)} + 3 \left(\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{2 \cos{\left(x \right)}}{x^{2}}\right) \sin{\left(\frac{\cos{\left(x \right)}}{x} \right)} + \frac{\left(\sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}\right)^{2} \cos{\left(\frac{\cos{\left(x \right)}}{x} \right)}}{x}\right) \cos{\left(x \right)} + \frac{3 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(\sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}\right) \sin{\left(\frac{\cos{\left(x \right)}}{x} \right)}}{x} + \frac{\left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\left(\sin{\left(x \right)} + \frac{3 \cos{\left(x \right)}}{x} - \frac{6 \sin{\left(x \right)}}{x^{2}} - \frac{6 \cos{\left(x \right)}}{x^{3}}\right) \sin{\left(\frac{\cos{\left(x \right)}}{x} \right)} - \frac{3 \left(\sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}\right) \left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{2 \cos{\left(x \right)}}{x^{2}}\right) \cos{\left(\frac{\cos{\left(x \right)}}{x} \right)}}{x} + \frac{\left(\sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}\right)^{3} \sin{\left(\frac{\cos{\left(x \right)}}{x} \right)}}{x^{2}}\right)}{x} - \frac{6 \left(\left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \cos{\left(\frac{\cos{\left(x \right)}}{x} \right)} - 2 \left(\sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}\right) \sin{\left(\frac{\cos{\left(x \right)}}{x} \right)} \cos{\left(x \right)} + \frac{\left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\left(- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{2 \cos{\left(x \right)}}{x^{2}}\right) \sin{\left(\frac{\cos{\left(x \right)}}{x} \right)} + \frac{\left(\sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}\right)^{2} \cos{\left(\frac{\cos{\left(x \right)}}{x} \right)}}{x}\right)}{x}\right)}{x} - \frac{18 \left(x \cos{\left(x \right)} \cos{\left(\frac{\cos{\left(x \right)}}{x} \right)} + \frac{\left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}\right) \sin{\left(\frac{\cos{\left(x \right)}}{x} \right)}}{x}\right)}{x^{2}} + \frac{24 \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \cos{\left(\frac{\cos{\left(x \right)}}{x} \right)}}{x^{3}}}{x^{2}}

Simplificar

Gráfico

Derivada de -((xsin(x)+cos(x))cos(cosx/x))/x^2

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Derivada de -((xsin(x)+cos(x))cos(cosx/x))/x^2

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