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(е^-x):((sin^2)x)
Derivada de la función/ sin^2/ (е^-x):((sin^2)x)

Derivada de (е^-x):((sin^2)x)

Función f() - derivada -er orden en el punto
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Gráfico:

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Solución

Ha introducido [src] 
    -x   
   E     
---------
   2     
sin (x)*x
e−xxsin⁡2(x)\frac{e^{- x}}{x \sin^{2}{\left(x \right)}} 
E^(-x)/((sin(x)^2*x))
Solución detallada

  1. Se aplica la regla de la derivada parcial:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=1f{\left(x \right)} = 1 y g(x)=xexsin⁡2(x)g{\left(x \right)} = x e^{x} \sin^{2}{\left(x \right)}.

    Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. La derivada de una constante 11 es igual a cero.

    Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

      ddxf(x)g(x)h(x)=f(x)g(x)ddxh(x)+f(x)h(x)ddxg(x)+g(x)h(x)ddxf(x)\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(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)=sin⁡2(x)g{\left(x \right)} = \sin^{2}{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Sustituimos u=sin⁡(x)u = \sin{\left(x \right)}.

      2. Según el principio, aplicamos: u2u^{2} tenemos 2u2 u

      3. Luego se aplica una cadena de reglas. Multiplicamos por ddxsin⁡(x)\frac{d}{d x} \sin{\left(x \right)}:

        1. La derivada del seno es igual al coseno:

          ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

        Como resultado de la secuencia de reglas:

        2sin⁡(x)cos⁡(x)2 \sin{\left(x \right)} \cos{\left(x \right)}

      h(x)=exh{\left(x \right)} = e^{x}; calculamos ddxh(x)\frac{d}{d x} h{\left(x \right)}:

      1. Derivado exe^{x} es.

      Como resultado de: xexsin⁡2(x)+2xexsin⁡(x)cos⁡(x)+exsin⁡2(x)x e^{x} \sin^{2}{\left(x \right)} + 2 x e^{x} \sin{\left(x \right)} \cos{\left(x \right)} + e^{x} \sin^{2}{\left(x \right)}

    Ahora aplicamos la regla de la derivada de una divesión:

    (−xexsin⁡2(x)−2xexsin⁡(x)cos⁡(x)−exsin⁡2(x))e−2xx2sin⁡4(x)\frac{\left(- x e^{x} \sin^{2}{\left(x \right)} - 2 x e^{x} \sin{\left(x \right)} \cos{\left(x \right)} - e^{x} \sin^{2}{\left(x \right)}\right) e^{- 2 x}}{x^{2} \sin^{4}{\left(x \right)}}

  2. Simplificamos:

    −(x+2xtan⁡(x)+1)e−xx2sin⁡2(x)- \frac{\left(x + \frac{2 x}{\tan{\left(x \right)}} + 1\right) e^{- x}}{x^{2} \sin^{2}{\left(x \right)}}

Respuesta:

−(x+2xtan⁡(x)+1)e−xx2sin⁡2(x)- \frac{\left(x + \frac{2 x}{\tan{\left(x \right)}} + 1\right) e^{- x}}{x^{2} \sin^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-100000000100000000
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Primera derivada [src] 
                  /     2                       \  -x
      1      -x   \- sin (x) - 2*x*cos(x)*sin(x)/*e  
- ---------*e   + -----------------------------------
       2                        2    4               
  x*sin (x)                    x *sin (x)
−1xsin⁡2(x)e−x+(−2xsin⁡(x)cos⁡(x)−sin⁡2(x))e−xx2sin⁡4(x)- \frac{1}{x \sin^{2}{\left(x \right)}} e^{- x} + \frac{\left(- 2 x \sin{\left(x \right)} \cos{\left(x \right)} - \sin^{2}{\left(x \right)}\right) e^{- x}}{x^{2} \sin^{4}{\left(x \right)}}
Simplificar
Segunda derivada [src] 
/                                                                   /     2           2                     \                                                           \    
|    2*x*cos(x) + sin(x)   /1   2*cos(x)\                         2*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/   2*(2*x*cos(x) + sin(x))*cos(x)                          |    
|    ------------------- + |- + --------|*(2*x*cos(x) + sin(x)) - ------------------------------------------- + ------------------------------                          |    
|             x            \x    sin(x) /                                            sin(x)                                 sin(x)               2*(2*x*cos(x) + sin(x))|  -x
|1 + ----------------------------------------------------------------------------------------------------------------------------------------- + -----------------------|*e  
\                                                                     x*sin(x)                                                                           x*sin(x)       /    
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                       2                                                                                     
                                                                                  x*sin (x)
(1+2(2xcos⁡(x)+sin⁡(x))xsin⁡(x)+(2xcos⁡(x)+sin⁡(x))(2cos⁡(x)sin⁡(x)+1x)+2(2xcos⁡(x)+sin⁡(x))cos⁡(x)sin⁡(x)−2(−xsin⁡2(x)+xcos⁡2(x)+2sin⁡(x)cos⁡(x))sin⁡(x)+2xcos⁡(x)+sin⁡(x)xxsin⁡(x))e−xxsin⁡2(x)\frac{\left(1 + \frac{2 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right)}{x \sin{\left(x \right)}} + \frac{\left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{1}{x}\right) + \frac{2 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{2 \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right)}{\sin{\left(x \right)}} + \frac{2 x \cos{\left(x \right)} + \sin{\left(x \right)}}{x}}{x \sin{\left(x \right)}}\right) e^{- x}}{x \sin^{2}{\left(x \right)}}
Simplificar
Tercera derivada [src] 
 /                                                                                                                                                                  /1   2*cos(x)\                                                                                                                               /1   2*cos(x)\ /     2           2                     \                                        /1   2*cos(x)\                                                                                                                                                                                                                                        \     
 |                 /       2           2                       \                           /              2              \                                          |- + --------|*(2*x*cos(x) + sin(x))      /     2           2                     \            /     2           2                     \   2*|- + --------|*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/         2                            2*|- + --------|*(2*x*cos(x) + sin(x))*cos(x)                                                                                                                                                                                                           |     
 |               2*\- 3*cos (x) + 3*sin (x) + 4*x*cos(x)*sin(x)/                           |    1    3*cos (x)   2*cos(x)|   3*(2*x*cos(x) + sin(x))                \x    sin(x) /                         12*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/*cos(x)   6*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/     \x    sin(x) /                                             10*cos (x)*(2*x*cos(x) + sin(x))     \x    sin(x) /                                8*(2*x*cos(x) + sin(x))*cos(x)                               /                                                               /     2           2                     \                                 \|     
 |    2*sin(x) - ----------------------------------------------- + 2*(2*x*cos(x) + sin(x))*|1 + -- + --------- + --------| + ----------------------- + 4*x*cos(x) + ------------------------------------ - --------------------------------------------------- - ------------------------------------------- - ---------------------------------------------------------- + -------------------------------- + --------------------------------------------- + ------------------------------                               |2*x*cos(x) + sin(x)   /1   2*cos(x)\                         2*\x*cos (x) - x*sin (x) + 2*cos(x)*sin(x)/   2*(2*x*cos(x) + sin(x))*cos(x)||     
 |                                    sin(x)                                               |     2       2       x*sin(x)|               2                                           x                                              2                                              x*sin(x)                                              sin(x)                                            2                                       sin(x)                                 x*sin(x)                                        3*|------------------- + |- + --------|*(2*x*cos(x) + sin(x)) - ------------------------------------------- + ------------------------------||     
 |                                                                                         \    x     sin (x)            /              x                                                                                        sin (x)                                                                                                                                                sin (x)                                                                                                 3*(2*x*cos(x) + sin(x))     \         x            \x    sin(x) /                                            sin(x)                                 sin(x)            /|  -x 
-|1 + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------|*e   
 \                                                                                                                                                                                                                                            x*sin(x)                                                                                                                                                                                                                                                  x*sin(x)                                                                             x*sin(x)                                                                  /     
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                       2                                                                                                                                                                                                                                                                                                                                     
                                                                                                                                                                                                                                                                                                                                  x*sin (x)
−(1+3(2xcos⁡(x)+sin⁡(x))xsin⁡(x)+3((2xcos⁡(x)+sin⁡(x))(2cos⁡(x)sin⁡(x)+1x)+2(2xcos⁡(x)+sin⁡(x))cos⁡(x)sin⁡(x)−2(−xsin⁡2(x)+xcos⁡2(x)+2sin⁡(x)cos⁡(x))sin⁡(x)+2xcos⁡(x)+sin⁡(x)x)xsin⁡(x)+4xcos⁡(x)+2(2xcos⁡(x)+sin⁡(x))(2cos⁡(x)sin⁡(x)+1x)cos⁡(x)sin⁡(x)+2(2xcos⁡(x)+sin⁡(x))(1+3cos⁡2(x)sin⁡2(x)+2cos⁡(x)xsin⁡(x)+1x2)+10(2xcos⁡(x)+sin⁡(x))cos⁡2(x)sin⁡2(x)−2(2cos⁡(x)sin⁡(x)+1x)(−xsin⁡2(x)+xcos⁡2(x)+2sin⁡(x)cos⁡(x))sin⁡(x)−12(−xsin⁡2(x)+xcos⁡2(x)+2sin⁡(x)cos⁡(x))cos⁡(x)sin⁡2(x)−2(4xsin⁡(x)cos⁡(x)+3sin⁡2(x)−3cos⁡2(x))sin⁡(x)+2sin⁡(x)+(2xcos⁡(x)+sin⁡(x))(2cos⁡(x)sin⁡(x)+1x)x+8(2xcos⁡(x)+sin⁡(x))cos⁡(x)xsin⁡(x)−6(−xsin⁡2(x)+xcos⁡2(x)+2sin⁡(x)cos⁡(x))xsin⁡(x)+3(2xcos⁡(x)+sin⁡(x))x2xsin⁡(x))e−xxsin⁡2(x)- \frac{\left(1 + \frac{3 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right)}{x \sin{\left(x \right)}} + \frac{3 \left(\left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{1}{x}\right) + \frac{2 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{2 \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right)}{\sin{\left(x \right)}} + \frac{2 x \cos{\left(x \right)} + \sin{\left(x \right)}}{x}\right)}{x \sin{\left(x \right)}} + \frac{4 x \cos{\left(x \right)} + \frac{2 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{1}{x}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(1 + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \cos{\left(x \right)}}{x \sin{\left(x \right)}} + \frac{1}{x^{2}}\right) + \frac{10 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \left(\frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{1}{x}\right) \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right)}{\sin{\left(x \right)}} - \frac{12 \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \left(4 x \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right)}{\sin{\left(x \right)}} + 2 \sin{\left(x \right)} + \frac{\left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{1}{x}\right)}{x} + \frac{8 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \right)}}{x \sin{\left(x \right)}} - \frac{6 \left(- x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right)}{x \sin{\left(x \right)}} + \frac{3 \left(2 x \cos{\left(x \right)} + \sin{\left(x \right)}\right)}{x^{2}}}{x \sin{\left(x \right)}}\right) e^{- x}}{x \sin^{2}{\left(x \right)}}
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Gráfico
Derivada de (е^-x):((sin^2)x)
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