Derivada de (x(xsin(x)+2cos(x)))/(x(xcos(x)-2sin(x)))

Ha introducido [src]

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

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

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

Solución detallada

Se aplica la regla de la derivada parcial:

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

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

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

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

           1                                                             (x*sin(x) + 2*cos(x))*(2*sin(x) - x*(-cos(x) - x*sin(x)) - x*cos(x))
-----------------------*(2*cos(x) + x*(-sin(x) + x*cos(x)) + x*sin(x)) + --------------------------------------------------------------------
x*(x*cos(x) - 2*sin(x))                                                                                               2                      
                                                                                               x*(x*cos(x) - 2*sin(x))

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

Gráfico

Derivada de (x(xsin(x)+2cos(x)))/(x(xcos(x)-2sin(x)))

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Derivada de (x(xsin(x)+2cos(x)))/(x(xcos(x)-2sin(x)))

Gráfico:

Definida a trozos:

Solución