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y=cosx/(1-sinx)x*exp(-x)
Derivada de la función/ y=cos/ x*exp/ -sinx/ exp(-x)/ y=cosx/(1-sinx)/ y=cosx/(1-sinx)x*exp(-x)

Derivada de y=cosx/(1-sinx)x*exp(-x)

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

Ha introducido [src] 
  cos(x)      -x
----------*x*e  
1 - sin(x)
xcos⁡(x)1−sin⁡(x)e−xx \frac{\cos{\left(x \right)}}{1 - \sin{\left(x \right)}} e^{- x} 
((cos(x)/(1 - sin(x)))*x)*exp(-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)=xcos⁡(x)f{\left(x \right)} = x \cos{\left(x \right)} y g(x)=(1−sin⁡(x))exg{\left(x \right)} = \left(1 - \sin{\left(x \right)}\right) e^{x}.

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

    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)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. La derivada del coseno es igual a menos el seno:

        ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

      Como resultado de: −xsin⁡(x)+cos⁡(x)- x \sin{\left(x \right)} + \cos{\left(x \right)}

    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)=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)=1−sin⁡(x)f{\left(x \right)} = 1 - \sin{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. diferenciamos 1−sin⁡(x)1 - \sin{\left(x \right)} miembro por miembro:

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

        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:

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

          Entonces, como resultado: −cos⁡(x)- \cos{\left(x \right)}

        Como resultado de: −cos⁡(x)- \cos{\left(x \right)}

      g(x)=exg{\left(x \right)} = e^{x}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Derivado exe^{x} es.

      Como resultado de: (1−sin⁡(x))ex−excos⁡(x)\left(1 - \sin{\left(x \right)}\right) e^{x} - e^{x} \cos{\left(x \right)}

    Ahora aplicamos la regla de la derivada de una divesión:

    (−x((1−sin⁡(x))ex−excos⁡(x))cos⁡(x)+(1−sin⁡(x))(−xsin⁡(x)+cos⁡(x))ex)e−2x(1−sin⁡(x))2\frac{\left(- x \left(\left(1 - \sin{\left(x \right)}\right) e^{x} - e^{x} \cos{\left(x \right)}\right) \cos{\left(x \right)} + \left(1 - \sin{\left(x \right)}\right) \left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}\right) e^{- 2 x}}{\left(1 - \sin{\left(x \right)}\right)^{2}}

  2. Simplificamos:

    (−x(−2sin⁡(x+π4)+1)cos⁡(x)+(xsin⁡(x)−cos⁡(x))(sin⁡(x)−1))e−x(sin⁡(x)−1)2\frac{\left(- x \left(- \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} + 1\right) \cos{\left(x \right)} + \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(\sin{\left(x \right)} - 1\right)\right) e^{- x}}{\left(\sin{\left(x \right)} - 1\right)^{2}}

Respuesta:

(−x(−2sin⁡(x+π4)+1)cos⁡(x)+(xsin⁡(x)−cos⁡(x))(sin⁡(x)−1))e−x(sin⁡(x)−1)2\frac{\left(- x \left(- \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} + 1\right) \cos{\left(x \right)} + \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(\sin{\left(x \right)} - 1\right)\right) e^{- x}}{\left(\sin{\left(x \right)} - 1\right)^{2}}

Gráfica
02468-8-6-4-2-1010-10000001000000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
/  /      2                   \             \                 -x
|  |   cos (x)        sin(x)  |     cos(x)  |  -x   x*cos(x)*e  
|x*|------------- - ----------| + ----------|*e   - ------------
|  |            2   1 - sin(x)|   1 - sin(x)|        1 - sin(x) 
\  \(1 - sin(x))              /             /
−xe−xcos⁡(x)1−sin⁡(x)+(x(−sin⁡(x)1−sin⁡(x)+cos⁡2(x)(1−sin⁡(x))2)+cos⁡(x)1−sin⁡(x))e−x- \frac{x e^{- x} \cos{\left(x \right)}}{1 - \sin{\left(x \right)}} + \left(x \left(- \frac{\sin{\left(x \right)}}{1 - \sin{\left(x \right)}} + \frac{\cos^{2}{\left(x \right)}}{\left(1 - \sin{\left(x \right)}\right)^{2}}\right) + \frac{\cos{\left(x \right)}}{1 - \sin{\left(x \right)}}\right) e^{- x}
Simplificar
Segunda derivada [src] 
/                                                                              /           2                           \       \    
|                                                                              |      2*cos (x)                        |       |    
|                                     /     2              \         2         |     ----------- + sin(x)              |       |    
|                                     |  cos (x)           |    2*cos (x)      |     -1 + sin(x)              2*sin(x) |       |  -x
|2*cos(x) + 2*sin(x) - x*cos(x) - 2*x*|----------- + sin(x)| + ----------- - x*|-1 + -------------------- + -----------|*cos(x)|*e  
\                                     \-1 + sin(x)         /   -1 + sin(x)     \         -1 + sin(x)        -1 + sin(x)/       /    
------------------------------------------------------------------------------------------------------------------------------------
                                                            -1 + sin(x)
(−2x(sin⁡(x)+cos⁡2(x)sin⁡(x)−1)−x(−1+sin⁡(x)+2cos⁡2(x)sin⁡(x)−1sin⁡(x)−1+2sin⁡(x)sin⁡(x)−1)cos⁡(x)−xcos⁡(x)+2sin⁡(x)+2cos⁡(x)+2cos⁡2(x)sin⁡(x)−1)e−xsin⁡(x)−1\frac{\left(- 2 x \left(\sin{\left(x \right)} + \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}\right) - x \left(-1 + \frac{\sin{\left(x \right)} + \frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}}{\sin{\left(x \right)} - 1} + \frac{2 \sin{\left(x \right)}}{\sin{\left(x \right)} - 1}\right) \cos{\left(x \right)} - x \cos{\left(x \right)} + 2 \sin{\left(x \right)} + 2 \cos{\left(x \right)} + \frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}\right) e^{- x}}{\sin{\left(x \right)} - 1}
Simplificar
Tercera derivada [src] 
/                                    /                      /                          2      \                                           \                                                                                                                                                       \    
|                                    |                 2    |       6*sin(x)      6*cos (x)   |     /      2             \                |                   /           2                           \                                           /           2                           \       |    
|                                    |              cos (x)*|-1 + ----------- + --------------|     | 2*cos (x)          |                |                   |      2*cos (x)                        |                                           |      2*cos (x)                        |       |    
|                                    |      2               |     -1 + sin(x)                2|   3*|----------- + sin(x)|*sin(x)         |         2         |     ----------- + sin(x)              |              /     2              \       |     ----------- + sin(x)              |       |    
|                                    | 3*cos (x)            \                   (-1 + sin(x)) /     \-1 + sin(x)         /                |    6*cos (x)      |     -1 + sin(x)              2*sin(x) |              |  cos (x)           |       |     -1 + sin(x)              2*sin(x) |       |  -x
|-6*sin(x) - 3*cos(x) + x*cos(x) - x*|----------- - ------------------------------------------- - ------------------------------- + sin(x)| - ----------- - 3*|-1 + -------------------- + -----------|*cos(x) + 3*x*|----------- + sin(x)| + 3*x*|-1 + -------------------- + -----------|*cos(x)|*e  
\                                    \-1 + sin(x)                   -1 + sin(x)                             -1 + sin(x)                   /   -1 + sin(x)     \         -1 + sin(x)        -1 + sin(x)/              \-1 + sin(x)         /       \         -1 + sin(x)        -1 + sin(x)/       /    
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                              -1 + sin(x)
(3x(sin⁡(x)+cos⁡2(x)sin⁡(x)−1)+3x(−1+sin⁡(x)+2cos⁡2(x)sin⁡(x)−1sin⁡(x)−1+2sin⁡(x)sin⁡(x)−1)cos⁡(x)−x(sin⁡(x)−3(sin⁡(x)+2cos⁡2(x)sin⁡(x)−1)sin⁡(x)sin⁡(x)−1−(−1+6sin⁡(x)sin⁡(x)−1+6cos⁡2(x)(sin⁡(x)−1)2)cos⁡2(x)sin⁡(x)−1+3cos⁡2(x)sin⁡(x)−1)+xcos⁡(x)−3(−1+sin⁡(x)+2cos⁡2(x)sin⁡(x)−1sin⁡(x)−1+2sin⁡(x)sin⁡(x)−1)cos⁡(x)−6sin⁡(x)−3cos⁡(x)−6cos⁡2(x)sin⁡(x)−1)e−xsin⁡(x)−1\frac{\left(3 x \left(\sin{\left(x \right)} + \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}\right) + 3 x \left(-1 + \frac{\sin{\left(x \right)} + \frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}}{\sin{\left(x \right)} - 1} + \frac{2 \sin{\left(x \right)}}{\sin{\left(x \right)} - 1}\right) \cos{\left(x \right)} - x \left(\sin{\left(x \right)} - \frac{3 \left(\sin{\left(x \right)} + \frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{\sin{\left(x \right)} - 1} - \frac{\left(-1 + \frac{6 \sin{\left(x \right)}}{\sin{\left(x \right)} - 1} + \frac{6 \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right)^{2}}\right) \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1} + \frac{3 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}\right) + x \cos{\left(x \right)} - 3 \left(-1 + \frac{\sin{\left(x \right)} + \frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}}{\sin{\left(x \right)} - 1} + \frac{2 \sin{\left(x \right)}}{\sin{\left(x \right)} - 1}\right) \cos{\left(x \right)} - 6 \sin{\left(x \right)} - 3 \cos{\left(x \right)} - \frac{6 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}\right) e^{- x}}{\sin{\left(x \right)} - 1}
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Gráfico
Derivada de y=cosx/(1-sinx)x*exp(-x)
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