Derivada de (x-sinx)(x-pi)^2/(x^2sinx)

Ha introducido [src]

                     2
(x - sin(x))*(x - pi) 
----------------------
       2              
      x *sin(x)

\frac{\left(x - \pi\right)^{2} \left(x - \sin{\left(x \right)}\right)}{x^{2} \sin{\left(x \right)}}

((x - sin(x))*(x - pi)^2)/((x^2*sin(x)))

Solución detallada

Se aplica la regla de la derivada parcial:

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

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

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

$\frac{x^{2} \left(\left(1 - \cos{\left(x \right)}\right) \left(x - \pi\right)^{2} + \left(x - \sin{\left(x \right)}\right) \left(2 x - 2 \pi\right)\right) \sin{\left(x \right)} - \left(x - \pi\right)^{2} \left(x - \sin{\left(x \right)}\right) \left(x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}\right)}{x^{4} \sin^{2}{\left(x \right)}}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

                                                                                                                                                                       /                                                                                                                                                                                      /2   cos(x)\ /            2                    \     /2   cos(x)\                                                           /2   cos(x)\                                                                                                             \                                                                                                                                                                                          
                                                                                                                                                                       |                                            /              2              \                2                         /            2                    \                              |- + ------|*\2*sin(x) - x *sin(x) + 4*x*cos(x)/   2*|- + ------|*(2*sin(x) + x*cos(x))        2                            |- + ------|*(2*sin(x) + x*cos(x))*cos(x)     /            2                    \                                        |                                                                                                                                                                                          
                                                                                                                                                        2              |                                            |    6    2*cos (x)   4*cos(x)|   -6*cos(x) + x *cos(x) + 6*x*sin(x)   6*\2*sin(x) - x *sin(x) + 4*x*cos(x)/   10*(2*sin(x) + x*cos(x))   \x   sin(x)/                                         \x   sin(x)/                         3*cos (x)*(2*sin(x) + x*cos(x))   \x   sin(x)/                                3*\2*sin(x) - x *sin(x) + 4*x*cos(x)/*cos(x)   8*(2*sin(x) + x*cos(x))*cos(x)|                                                         /                                                 2                                                                             \
                                                                                                                                                (x - pi) *(x - sin(x))*|2*sin(x) + x*cos(x) + (2*sin(x) + x*cos(x))*|1 + -- + --------- + --------| - ---------------------------------- - ------------------------------------- + ------------------------ - ------------------------------------------------ + ------------------------------------ + ------------------------------- + ----------------------------------------- - -------------------------------------------- + ------------------------------|                                                         |/2   cos(x)\                         2*sin(x) - x *sin(x) + 4*x*cos(x)   2*(2*sin(x) + x*cos(x))   (2*sin(x) + x*cos(x))*cos(x)|
                                                                              /                          2                                  \                          |                                            |     2       2       x*sin(x)|                   x                                       2                                2                                     x                                            x                                    2                                    sin(x)                                      x*sin(x)                                x*sin(x)           |   3*(x - pi)*(-2*x + 2*sin(x) + (-1 + cos(x))*(x - pi))*||- + ------|*(2*sin(x) + x*cos(x)) - --------------------------------- + ----------------------- + ----------------------------|
                       2                              3*(2*sin(x) + x*cos(x))*\-2*sin(x) + 2*x + (x - pi) *sin(x) - 4*(-1 + cos(x))*(x - pi)/                          \                                            \    x     sin (x)            /                                                          x                                x                                                                                                                     sin (x)                                                                                                                                        /                                                         \\x   sin(x)/                                         x                              x                         sin(x)           /
6 - 6*cos(x) + (x - pi) *cos(x) + 6*(x - pi)*sin(x) - --------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                              x*sin(x)                                                                                                                                                                                                                                                                        x*sin(x)                                                                                                                                                                                                                                                                                                                         x*sin(x)                                                                                       
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                   2                                                                                                                                                                                                                                                                                                                                                                                                          
                                                                                                                                                                                                                                                                                                                                                                                                  x *sin(x)

\frac{\left(x - \pi\right)^{2} \cos{\left(x \right)} + 6 \left(x - \pi\right) \sin{\left(x \right)} - 6 \cos{\left(x \right)} + 6 - \frac{\left(x - \pi\right)^{2} \left(x - \sin{\left(x \right)}\right) \left(x \cos{\left(x \right)} + \frac{\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2}{x}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{4 \cos{\left(x \right)}}{x \sin{\left(x \right)}} + \frac{6}{x^{2}}\right) + \frac{3 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 2 \sin{\left(x \right)} + \frac{2 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2}{x}\right)}{x} + \frac{8 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos{\left(x \right)}}{x \sin{\left(x \right)}} - \frac{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2}{x}\right) \left(- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x} - \frac{3 \left(- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos{\left(x \right)}}{x \sin{\left(x \right)}} - \frac{x^{2} \cos{\left(x \right)} + 6 x \sin{\left(x \right)} - 6 \cos{\left(x \right)}}{x} + \frac{10 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x^{2}} - \frac{6 \left(- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x^{2}}\right)}{x \sin{\left(x \right)}} - \frac{3 \left(x - \pi\right) \left(- 2 x + \left(x - \pi\right) \left(\cos{\left(x \right)} - 1\right) + 2 \sin{\left(x \right)}\right) \left(\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2}{x}\right) + \frac{\left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)}{x} - \frac{- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}}{x}\right)}{x \sin{\left(x \right)}} - \frac{3 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(2 x + \left(x - \pi\right)^{2} \sin{\left(x \right)} - 4 \left(x - \pi\right) \left(\cos{\left(x \right)} - 1\right) - 2 \sin{\left(x \right)}\right)}{x \sin{\left(x \right)}}}{x^{2} \sin{\left(x \right)}}

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Gráfico

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