Derivada de y=sinxln(x+cosx)x*exp(-x)

Solución

Ha introducido [src]

                          -x
sin(x)*log(x + cos(x))*x*e

$$x \log{\left(x + \cos{\left(x \right)} \right)} \sin{\left(x \right)} e^{- x}$$

((sin(x)*log(x + cos(x)))*x)*exp(-x)

Solución detallada

Se aplica la regla de la derivada parcial:

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

$f{\left(x \right)} = x \log{\left(x + \cos{\left(x \right)} \right)} \sin{\left(x \right)}$ y $g{\left(x \right)} = e^{x}$ .

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

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

$\frac{\left(- x \left(x + \cos{\left(x \right)}\right) \log{\left(x + \cos{\left(x \right)} \right)} \sin{\left(x \right)} - x \left(\sin{\left(x \right)} - 1\right) \sin{\left(x \right)} + \left(x + \cos{\left(x \right)}\right) \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x + \cos{\left(x \right)} \right)}\right) e^{- x}}{x + \cos{\left(x \right)}}$

Respuesta:

$\frac{\left(- x \left(x + \cos{\left(x \right)}\right) \log{\left(x + \cos{\left(x \right)} \right)} \sin{\left(x \right)} - x \left(\sin{\left(x \right)} - 1\right) \sin{\left(x \right)} + \left(x + \cos{\left(x \right)}\right) \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x + \cos{\left(x \right)} \right)}\right) e^{- x}}{x + \cos{\left(x \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

$$- x e^{- x} \log{\left(x + \cos{\left(x \right)} \right)} \sin{\left(x \right)} + \left(x \left(\frac{\left(1 - \sin{\left(x \right)}\right) \sin{\left(x \right)}}{x + \cos{\left(x \right)}} + \log{\left(x + \cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) + \log{\left(x + \cos{\left(x \right)} \right)} \sin{\left(x \right)}\right) e^{- x}$$

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

/    /                         /                         3                         \                                                                     \                                                                                                                                                                                                                                                                                              \    
|    |                         |          2*(-1 + sin(x))    3*(-1 + sin(x))*cos(x)|                                     /             2         \       |                                                                                        /                         /             2         \                                \                                                         /             2         \                                |    
|    |                         |-sin(x) + ---------------- + ----------------------|*sin(x)                              |(-1 + sin(x))          |       |                                                                                        |                         |(-1 + sin(x))          |                                |                                                         |(-1 + sin(x))          |                                |    
|    |                         |                       2           x + cos(x)      |                                   3*|-------------- + cos(x)|*cos(x)|                                                                                        |                         |-------------- + cos(x)|*sin(x)                         |                                                       3*|-------------- + cos(x)|*sin(x)                         |    
|    |                         \           (x + cos(x))                            /          3*(-1 + sin(x))*sin(x)     \  x + cos(x)           /       |                                  /                         (-1 + sin(x))*sin(x)\       |                         \  x + cos(x)           /          2*(-1 + sin(x))*cos(x)|                              6*(-1 + sin(x))*cos(x)     \  x + cos(x)           /          6*(-1 + sin(x))*sin(x)|  -x
|- x*|cos(x)*log(x + cos(x)) + ------------------------------------------------------------ - ---------------------- + ----------------------------------| - 6*cos(x)*log(x + cos(x)) + 3*x*|cos(x)*log(x + cos(x)) - --------------------| + 3*x*|log(x + cos(x))*sin(x) + -------------------------------- + ----------------------| - x*log(x + cos(x))*sin(x) - ---------------------- - ---------------------------------- + ----------------------|*e  
\    \                                                  x + cos(x)                                  x + cos(x)                     x + cos(x)            /                                  \                              x + cos(x)     /       \                                    x + cos(x)                    x + cos(x)      /                                    x + cos(x)                     x + cos(x)                     x + cos(x)      /

$$\left(3 x \left(\log{\left(x + \cos{\left(x \right)} \right)} \cos{\left(x \right)} - \frac{\left(\sin{\left(x \right)} - 1\right) \sin{\left(x \right)}}{x + \cos{\left(x \right)}}\right) + 3 x \left(\log{\left(x + \cos{\left(x \right)} \right)} \sin{\left(x \right)} + \frac{2 \left(\sin{\left(x \right)} - 1\right) \cos{\left(x \right)}}{x + \cos{\left(x \right)}} + \frac{\left(\cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} - 1\right)^{2}}{x + \cos{\left(x \right)}}\right) \sin{\left(x \right)}}{x + \cos{\left(x \right)}}\right) - x \left(\log{\left(x + \cos{\left(x \right)} \right)} \cos{\left(x \right)} - \frac{3 \left(\sin{\left(x \right)} - 1\right) \sin{\left(x \right)}}{x + \cos{\left(x \right)}} + \frac{3 \left(\cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} - 1\right)^{2}}{x + \cos{\left(x \right)}}\right) \cos{\left(x \right)}}{x + \cos{\left(x \right)}} + \frac{\left(- \sin{\left(x \right)} + \frac{3 \left(\sin{\left(x \right)} - 1\right) \cos{\left(x \right)}}{x + \cos{\left(x \right)}} + \frac{2 \left(\sin{\left(x \right)} - 1\right)^{3}}{\left(x + \cos{\left(x \right)}\right)^{2}}\right) \sin{\left(x \right)}}{x + \cos{\left(x \right)}}\right) - x \log{\left(x + \cos{\left(x \right)} \right)} \sin{\left(x \right)} - 6 \log{\left(x + \cos{\left(x \right)} \right)} \cos{\left(x \right)} + \frac{6 \left(\sin{\left(x \right)} - 1\right) \sin{\left(x \right)}}{x + \cos{\left(x \right)}} - \frac{6 \left(\sin{\left(x \right)} - 1\right) \cos{\left(x \right)}}{x + \cos{\left(x \right)}} - \frac{3 \left(\cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} - 1\right)^{2}}{x + \cos{\left(x \right)}}\right) \sin{\left(x \right)}}{x + \cos{\left(x \right)}}\right) e^{- x}$$

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Derivada de y=sinxln(x+cosx)x*exp(-x)

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