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Derivada de la función/ x*exp(-(-33cos(t)*80sin(2t)+33sin(t)*40cos(2t))/(1089cos^2(t)+1600cos^2(2t))x)

Derivada de x*exp(-(-33cos(t)*80sin(2t)+33sin(t)*40cos(2t))/(1089cos^2(t)+1600cos^2(2t))x)

Función f() - derivada -er orden en el punto
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Solución

Ha introducido [src] 
   --33*cos(t)*80*sin(2*t) - 33*sin(t)*40*cos(2*t)  
   -----------------------------------------------*x
                    2              2                
            1089*cos (t) + 1600*cos (2*t)           
x*e
xex−40⋅33sin⁡(t)cos⁡(2t)−80(−33cos⁡(t))sin⁡(2t)1089cos⁡2(t)+1600cos⁡2(2t)x e^{x \frac{- 40 \cdot 33 \sin{\left(t \right)} \cos{\left(2 t \right)} - 80 \left(- 33 \cos{\left(t \right)}\right) \sin{\left(2 t \right)}}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}}} 
x*exp(((-(-33*cos(t))*80*sin(2*t) - (33*sin(t))*40*cos(2*t))/(1089*cos(t)^2 + 1600*cos(2*t)^2))*x)
Solución detallada

  1. Se aplica la regla de la derivada de una multiplicación:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=xf{\left(x \right)} = x; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Según el principio, aplicamos: xx tenemos 11

    g(x)=ex−40⋅33sin⁡(t)cos⁡(2t)−80(−33cos⁡(t))sin⁡(2t)1089cos⁡2(t)+1600cos⁡2(2t)g{\left(x \right)} = e^{x \frac{- 40 \cdot 33 \sin{\left(t \right)} \cos{\left(2 t \right)} - 80 \left(- 33 \cos{\left(t \right)}\right) \sin{\left(2 t \right)}}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}}}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Sustituimos u=x−40⋅33sin⁡(t)cos⁡(2t)−80(−33cos⁡(t))sin⁡(2t)1089cos⁡2(t)+1600cos⁡2(2t)u = x \frac{- 40 \cdot 33 \sin{\left(t \right)} \cos{\left(2 t \right)} - 80 \left(- 33 \cos{\left(t \right)}\right) \sin{\left(2 t \right)}}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}}.

    2. Derivado eue^{u} es.

    3. Luego se aplica una cadena de reglas. Multiplicamos por ∂∂xx−40⋅33sin⁡(t)cos⁡(2t)−80(−33cos⁡(t))sin⁡(2t)1089cos⁡2(t)+1600cos⁡2(2t)\frac{\partial}{\partial x} x \frac{- 40 \cdot 33 \sin{\left(t \right)} \cos{\left(2 t \right)} - 80 \left(- 33 \cos{\left(t \right)}\right) \sin{\left(2 t \right)}}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \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. Según el principio, aplicamos: xx tenemos 11

        Entonces, como resultado: −40⋅33sin⁡(t)cos⁡(2t)−80(−33cos⁡(t))sin⁡(2t)1089cos⁡2(t)+1600cos⁡2(2t)\frac{- 40 \cdot 33 \sin{\left(t \right)} \cos{\left(2 t \right)} - 80 \left(- 33 \cos{\left(t \right)}\right) \sin{\left(2 t \right)}}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}}

      Como resultado de la secuencia de reglas:

      (−40⋅33sin⁡(t)cos⁡(2t)−80(−33cos⁡(t))sin⁡(2t))ex−40⋅33sin⁡(t)cos⁡(2t)−80(−33cos⁡(t))sin⁡(2t)1089cos⁡2(t)+1600cos⁡2(2t)1089cos⁡2(t)+1600cos⁡2(2t)\frac{\left(- 40 \cdot 33 \sin{\left(t \right)} \cos{\left(2 t \right)} - 80 \left(- 33 \cos{\left(t \right)}\right) \sin{\left(2 t \right)}\right) e^{x \frac{- 40 \cdot 33 \sin{\left(t \right)} \cos{\left(2 t \right)} - 80 \left(- 33 \cos{\left(t \right)}\right) \sin{\left(2 t \right)}}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}}}}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}}

    Como resultado de: x(−40⋅33sin⁡(t)cos⁡(2t)−80(−33cos⁡(t))sin⁡(2t))ex−40⋅33sin⁡(t)cos⁡(2t)−80(−33cos⁡(t))sin⁡(2t)1089cos⁡2(t)+1600cos⁡2(2t)1089cos⁡2(t)+1600cos⁡2(2t)+ex−40⋅33sin⁡(t)cos⁡(2t)−80(−33cos⁡(t))sin⁡(2t)1089cos⁡2(t)+1600cos⁡2(2t)\frac{x \left(- 40 \cdot 33 \sin{\left(t \right)} \cos{\left(2 t \right)} - 80 \left(- 33 \cos{\left(t \right)}\right) \sin{\left(2 t \right)}\right) e^{x \frac{- 40 \cdot 33 \sin{\left(t \right)} \cos{\left(2 t \right)} - 80 \left(- 33 \cos{\left(t \right)}\right) \sin{\left(2 t \right)}}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}}}}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}} + e^{x \frac{- 40 \cdot 33 \sin{\left(t \right)} \cos{\left(2 t \right)} - 80 \left(- 33 \cos{\left(t \right)}\right) \sin{\left(2 t \right)}}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}}}

  2. Simplificamos:

    (−2640xsin⁡3(t)+3960xsin⁡(t)+6400sin⁡4(t)−7489sin⁡2(t)+2689)e−1320x(2sin⁡2(t)−3)sin⁡(t)6400sin⁡4(t)−7489sin⁡2(t)+26896400sin⁡4(t)−7489sin⁡2(t)+2689\frac{\left(- 2640 x \sin^{3}{\left(t \right)} + 3960 x \sin{\left(t \right)} + 6400 \sin^{4}{\left(t \right)} - 7489 \sin^{2}{\left(t \right)} + 2689\right) e^{- \frac{1320 x \left(2 \sin^{2}{\left(t \right)} - 3\right) \sin{\left(t \right)}}{6400 \sin^{4}{\left(t \right)} - 7489 \sin^{2}{\left(t \right)} + 2689}}}{6400 \sin^{4}{\left(t \right)} - 7489 \sin^{2}{\left(t \right)} + 2689}

Respuesta:

(−2640xsin⁡3(t)+3960xsin⁡(t)+6400sin⁡4(t)−7489sin⁡2(t)+2689)e−1320x(2sin⁡2(t)−3)sin⁡(t)6400sin⁡4(t)−7489sin⁡2(t)+26896400sin⁡4(t)−7489sin⁡2(t)+2689\frac{\left(- 2640 x \sin^{3}{\left(t \right)} + 3960 x \sin{\left(t \right)} + 6400 \sin^{4}{\left(t \right)} - 7489 \sin^{2}{\left(t \right)} + 2689\right) e^{- \frac{1320 x \left(2 \sin^{2}{\left(t \right)} - 3\right) \sin{\left(t \right)}}{6400 \sin^{4}{\left(t \right)} - 7489 \sin^{2}{\left(t \right)} + 2689}}}{6400 \sin^{4}{\left(t \right)} - 7489 \sin^{2}{\left(t \right)} + 2689}

Primera derivada [src] 
                                                     --33*cos(t)*80*sin(2*t) - 33*sin(t)*40*cos(2*t)                                                       
                                                     -----------------------------------------------*x    --33*cos(t)*80*sin(2*t) - 33*sin(t)*40*cos(2*t)  
                                                                      2              2                    -----------------------------------------------*x
                                                              1089*cos (t) + 1600*cos (2*t)                                2              2                
x*(--33*cos(t)*80*sin(2*t) - 33*sin(t)*40*cos(2*t))*e                                                              1089*cos (t) + 1600*cos (2*t)           
------------------------------------------------------------------------------------------------------ + e                                                 
                                            2              2                                                                                               
                                    1089*cos (t) + 1600*cos (2*t)
x(−40⋅33sin⁡(t)cos⁡(2t)−80(−33cos⁡(t))sin⁡(2t))ex−40⋅33sin⁡(t)cos⁡(2t)−80(−33cos⁡(t))sin⁡(2t)1089cos⁡2(t)+1600cos⁡2(2t)1089cos⁡2(t)+1600cos⁡2(2t)+ex−40⋅33sin⁡(t)cos⁡(2t)−80(−33cos⁡(t))sin⁡(2t)1089cos⁡2(t)+1600cos⁡2(2t)\frac{x \left(- 40 \cdot 33 \sin{\left(t \right)} \cos{\left(2 t \right)} - 80 \left(- 33 \cos{\left(t \right)}\right) \sin{\left(2 t \right)}\right) e^{x \frac{- 40 \cdot 33 \sin{\left(t \right)} \cos{\left(2 t \right)} - 80 \left(- 33 \cos{\left(t \right)}\right) \sin{\left(2 t \right)}}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}}}}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}} + e^{x \frac{- 40 \cdot 33 \sin{\left(t \right)} \cos{\left(2 t \right)} - 80 \left(- 33 \cos{\left(t \right)}\right) \sin{\left(2 t \right)}}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}}}
Simplificar
Segunda derivada [src] 
                                                                                               -1320*x*(cos(2*t)*sin(t) - 2*cos(t)*sin(2*t))
                                                                                               ---------------------------------------------
                                                                                                               2              2             
     /     660*x*(cos(2*t)*sin(t) - 2*cos(t)*sin(2*t))\                                                1089*cos (t) + 1600*cos (2*t)        
2640*|-1 + -------------------------------------------|*(cos(2*t)*sin(t) - 2*cos(t)*sin(2*t))*e                                             
     |                    2              2            |                                                                                     
     \            1089*cos (t) + 1600*cos (2*t)       /                                                                                     
--------------------------------------------------------------------------------------------------------------------------------------------
                                                               2              2                                                             
                                                       1089*cos (t) + 1600*cos (2*t)
2640(sin⁡(t)cos⁡(2t)−2sin⁡(2t)cos⁡(t))(660x(sin⁡(t)cos⁡(2t)−2sin⁡(2t)cos⁡(t))1089cos⁡2(t)+1600cos⁡2(2t)−1)e−1320x(sin⁡(t)cos⁡(2t)−2sin⁡(2t)cos⁡(t))1089cos⁡2(t)+1600cos⁡2(2t)1089cos⁡2(t)+1600cos⁡2(2t)\frac{2640 \left(\sin{\left(t \right)} \cos{\left(2 t \right)} - 2 \sin{\left(2 t \right)} \cos{\left(t \right)}\right) \left(\frac{660 x \left(\sin{\left(t \right)} \cos{\left(2 t \right)} - 2 \sin{\left(2 t \right)} \cos{\left(t \right)}\right)}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}} - 1\right) e^{- \frac{1320 x \left(\sin{\left(t \right)} \cos{\left(2 t \right)} - 2 \sin{\left(2 t \right)} \cos{\left(t \right)}\right)}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}}}}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}}
Simplificar
Tercera derivada [src] 
                                                                                                  -1320*x*(cos(2*t)*sin(t) - 2*cos(t)*sin(2*t))
                                                                                                  ---------------------------------------------
                                                                                                                  2              2             
                                             2 /    440*x*(cos(2*t)*sin(t) - 2*cos(t)*sin(2*t))\          1089*cos (t) + 1600*cos (2*t)        
5227200*(cos(2*t)*sin(t) - 2*cos(t)*sin(2*t)) *|1 - -------------------------------------------|*e                                             
                                               |                   2              2            |                                               
                                               \           1089*cos (t) + 1600*cos (2*t)       /                                               
-----------------------------------------------------------------------------------------------------------------------------------------------
                                                                                       2                                                       
                                                        /        2              2     \                                                        
                                                        \1089*cos (t) + 1600*cos (2*t)/
5227200(sin⁡(t)cos⁡(2t)−2sin⁡(2t)cos⁡(t))2(−440x(sin⁡(t)cos⁡(2t)−2sin⁡(2t)cos⁡(t))1089cos⁡2(t)+1600cos⁡2(2t)+1)e−1320x(sin⁡(t)cos⁡(2t)−2sin⁡(2t)cos⁡(t))1089cos⁡2(t)+1600cos⁡2(2t)(1089cos⁡2(t)+1600cos⁡2(2t))2\frac{5227200 \left(\sin{\left(t \right)} \cos{\left(2 t \right)} - 2 \sin{\left(2 t \right)} \cos{\left(t \right)}\right)^{2} \left(- \frac{440 x \left(\sin{\left(t \right)} \cos{\left(2 t \right)} - 2 \sin{\left(2 t \right)} \cos{\left(t \right)}\right)}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}} + 1\right) e^{- \frac{1320 x \left(\sin{\left(t \right)} \cos{\left(2 t \right)} - 2 \sin{\left(2 t \right)} \cos{\left(t \right)}\right)}{1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}}}}{\left(1089 \cos^{2}{\left(t \right)} + 1600 \cos^{2}{\left(2 t \right)}\right)^{2}}
Simplificar
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