Derivada de x*sqrt(exp(-x)*cos(3x))exp(-x)

Ha introducido [src]

     ______________    
    /  -x            -x
x*\/  e  *cos(3*x) *e

x \sqrt{e^{- x} \cos{\left(3 x \right)}} e^{- x}

(x*sqrt(exp(-x)*cos(3*x)))*exp(-x)

Solución detallada

Se aplica la regla de la derivada parcial:

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

$f{\left(x \right)} = x \sqrt{\cos{\left(3 x \right)}}$ y $g{\left(x \right)} = e^{\frac{3 x}{2}}$ .

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

$\left(- \frac{3 x e^{\frac{3 x}{2}} \sqrt{\cos{\left(3 x \right)}}}{2} + \left(- \frac{3 x \sin{\left(3 x \right)}}{2 \sqrt{\cos{\left(3 x \right)}}} + \sqrt{\cos{\left(3 x \right)}}\right) e^{\frac{3 x}{2}}\right) e^{- 3 x}$
Simplificamos:

$\frac{\left(- \frac{3 \sqrt{2} x \sin{\left(3 x + \frac{\pi}{4} \right)}}{2} + \cos{\left(3 x \right)}\right) e^{- \frac{3 x}{2}}}{\sqrt{\cos{\left(3 x \right)}}}$

Respuesta:

$\frac{\left(- \frac{3 \sqrt{2} x \sin{\left(3 x + \frac{\pi}{4} \right)}}{2} + \cos{\left(3 x \right)}\right) e^{- \frac{3 x}{2}}}{\sqrt{\cos{\left(3 x \right)}}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

/                                                         x\                              
|                      /     -x                      -x\  -|                              
|                      |  3*e  *sin(3*x)   cos(3*x)*e  |  2|                       -x     
|   ______________   x*|- -------------- - ------------|*e |                       ---    
|  /  -x               \        2               2      /   |  -x       __________   2   -x
|\/  e  *cos(3*x)  + --------------------------------------|*e   - x*\/ cos(3*x) *e   *e  
|                                   __________             |                              
\                                 \/ cos(3*x)              /

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

Simplificar

Segunda derivada [src]

/                                                              /                                                             /  __________    3*sin(3*x) \                                                             \                            \      
|                                                              |                                                             |\/ cos(3*x)  + ------------|*(3*sin(3*x) + cos(3*x))                                     |                            |      
|                                                              |                                                             |                 __________|                                                             |                            |      
|                                                              |  2*(3*sin(3*x) + cos(3*x))   4*(-4*cos(3*x) + 3*sin(3*x))   \               \/ cos(3*x) /                           6*(3*sin(3*x) + cos(3*x))*sin(3*x)|                            |      
|                                                            x*|- ------------------------- + ---------------------------- + ----------------------------------------------------- - ----------------------------------|                            |  -3*x
|                                                              |           __________                   __________                                  cos(3*x)                                       3/2                 |                            |  ----
|      __________       __________   3*sin(3*x) + cos(3*x)     \         \/ cos(3*x)                  \/ cos(3*x)                                                                               cos   (3*x)            /   x*(3*sin(3*x) + cos(3*x))|   2  
|- 2*\/ cos(3*x)  + x*\/ cos(3*x)  - --------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------|*e    
|                                           __________                                                                                    4                                                                                         __________      |      
\                                         \/ cos(3*x)                                                                                                                                                                             \/ cos(3*x)       /

\left(\frac{x \left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right)}{\sqrt{\cos{\left(3 x \right)}}} + \frac{x \left(\frac{\left(\frac{3 \sin{\left(3 x \right)}}{\sqrt{\cos{\left(3 x \right)}}} + \sqrt{\cos{\left(3 x \right)}}\right) \left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right)}{\cos{\left(3 x \right)}} + \frac{4 \left(3 \sin{\left(3 x \right)} - 4 \cos{\left(3 x \right)}\right)}{\sqrt{\cos{\left(3 x \right)}}} - \frac{6 \left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) \sin{\left(3 x \right)}}{\cos^{\frac{3}{2}}{\left(3 x \right)}} - \frac{2 \left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right)}{\sqrt{\cos{\left(3 x \right)}}}\right)}{4} + x \sqrt{\cos{\left(3 x \right)}} - \frac{3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}}{\sqrt{\cos{\left(3 x \right)}}} - 2 \sqrt{\cos{\left(3 x \right)}}\right) e^{- \frac{3 x}{2}}

Simplificar

Tercera derivada [src]

/                                                                                                                                                                                                                                   /                                                                                                                      /                                      2     \                                                                                                                                                                                                                                                                                                                \                                                                                                                                                         \      
|                                                                     /                                                             /  __________    3*sin(3*x) \                                                             \     |                                                                                                                      |     __________    6*sin(3*x)    9*sin (3*x)|                                                                                  /  __________    3*sin(3*x) \                                /  __________    3*sin(3*x) \                                                                      /  __________    3*sin(3*x) \                                 |                                                                                                                                                         |      
|                                                                     |                                                             |\/ cos(3*x)  + ------------|*(3*sin(3*x) + cos(3*x))                                     |     |                                                                                              (3*sin(3*x) + cos(3*x))*|17*\/ cos(3*x)  - ------------ + -----------|                                                                                4*|\/ cos(3*x)  + ------------|*(-4*cos(3*x) + 3*sin(3*x))   2*|\/ cos(3*x)  + ------------|*(3*sin(3*x) + cos(3*x))                                            6*|\/ cos(3*x)  + ------------|*(3*sin(3*x) + cos(3*x))*sin(3*x)|                                                                                                                                                         |      
|                                                                     |                                                             |                 __________|                                                             |     |                                                                                                                      |                    __________      3/2     |         2                                                                        |                 __________|                                |                 __________|                                                                      |                 __________|                                 |                                                                                                                                                         |      
|                                                                     |  2*(3*sin(3*x) + cos(3*x))   4*(-4*cos(3*x) + 3*sin(3*x))   \               \/ cos(3*x) /                           6*(3*sin(3*x) + cos(3*x))*sin(3*x)|     |  38*(3*sin(3*x) + cos(3*x))   8*(9*sin(3*x) + 13*cos(3*x))   12*(-4*cos(3*x) + 3*sin(3*x))                           \                  \/ cos(3*x)    cos   (3*x)/   54*sin (3*x)*(3*sin(3*x) + cos(3*x))   12*(3*sin(3*x) + cos(3*x))*sin(3*x)     \               \/ cos(3*x) /                                \               \/ cos(3*x) /                           36*(-4*cos(3*x) + 3*sin(3*x))*sin(3*x)     \               \/ cos(3*x) /                                 |                                                                                                    /  __________    3*sin(3*x) \                        |      
|                                                                 3*x*|- ------------------------- + ---------------------------- + ----------------------------------------------------- - ----------------------------------|   x*|- -------------------------- + ---------------------------- + ----------------------------- + ---------------------------------------------------------------------- - ------------------------------------ - ----------------------------------- - ---------------------------------------------------------- + ------------------------------------------------------- + -------------------------------------- + ----------------------------------------------------------------|                                                                                                  3*|\/ cos(3*x)  + ------------|*(3*sin(3*x) + cos(3*x))|  -3*x
|                                                                     |           __________                   __________                                  cos(3*x)                                       3/2                 |     |           __________                    __________                      __________                                          cos(3*x)                                                 5/2                                    3/2                                             cos(3*x)                                                    cos(3*x)                                          3/2                                                    2                                 |                                                                                                    |                 __________|                        |  ----
|    __________       __________   3*(-4*cos(3*x) + 3*sin(3*x))       \         \/ cos(3*x)                  \/ cos(3*x)                                                                               cos   (3*x)            /     \         \/ cos(3*x)                   \/ cos(3*x)                     \/ cos(3*x)                                                                                                 cos   (3*x)                            cos   (3*x)                                                                                                                                                   cos   (3*x)                                            cos (3*x)                            /   3*(3*sin(3*x) + cos(3*x))   9*(3*sin(3*x) + cos(3*x))*sin(3*x)   3*x*(3*sin(3*x) + cos(3*x))     \               \/ cos(3*x) /                        |   2  
|3*\/ cos(3*x)  - x*\/ cos(3*x)  + ---------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------- - ---------------------------------- - --------------------------- + -------------------------------------------------------|*e    
|                                            __________                                                                                         4                                                                                                                                                                                                                                                                                                                            8                                                                                                                                                                                                                                                         __________                       3/2                              __________                                4*cos(3*x)                      |      
\                                          \/ cos(3*x)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             2*\/ cos(3*x)                   2*cos   (3*x)                     2*\/ cos(3*x)                                                                 /

\left(- \frac{3 x \left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right)}{2 \sqrt{\cos{\left(3 x \right)}}} - \frac{3 x \left(\frac{\left(\frac{3 \sin{\left(3 x \right)}}{\sqrt{\cos{\left(3 x \right)}}} + \sqrt{\cos{\left(3 x \right)}}\right) \left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right)}{\cos{\left(3 x \right)}} + \frac{4 \left(3 \sin{\left(3 x \right)} - 4 \cos{\left(3 x \right)}\right)}{\sqrt{\cos{\left(3 x \right)}}} - \frac{6 \left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) \sin{\left(3 x \right)}}{\cos^{\frac{3}{2}}{\left(3 x \right)}} - \frac{2 \left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right)}{\sqrt{\cos{\left(3 x \right)}}}\right)}{4} + \frac{x \left(- \frac{4 \left(\frac{3 \sin{\left(3 x \right)}}{\sqrt{\cos{\left(3 x \right)}}} + \sqrt{\cos{\left(3 x \right)}}\right) \left(3 \sin{\left(3 x \right)} - 4 \cos{\left(3 x \right)}\right)}{\cos{\left(3 x \right)}} + \frac{6 \left(\frac{3 \sin{\left(3 x \right)}}{\sqrt{\cos{\left(3 x \right)}}} + \sqrt{\cos{\left(3 x \right)}}\right) \left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) \sin{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + \frac{2 \left(\frac{3 \sin{\left(3 x \right)}}{\sqrt{\cos{\left(3 x \right)}}} + \sqrt{\cos{\left(3 x \right)}}\right) \left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right)}{\cos{\left(3 x \right)}} + \frac{36 \left(3 \sin{\left(3 x \right)} - 4 \cos{\left(3 x \right)}\right) \sin{\left(3 x \right)}}{\cos^{\frac{3}{2}}{\left(3 x \right)}} + \frac{12 \left(3 \sin{\left(3 x \right)} - 4 \cos{\left(3 x \right)}\right)}{\sqrt{\cos{\left(3 x \right)}}} + \frac{\left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) \left(\frac{9 \sin^{2}{\left(3 x \right)}}{\cos^{\frac{3}{2}}{\left(3 x \right)}} - \frac{6 \sin{\left(3 x \right)}}{\sqrt{\cos{\left(3 x \right)}}} + 17 \sqrt{\cos{\left(3 x \right)}}\right)}{\cos{\left(3 x \right)}} - \frac{54 \left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) \sin^{2}{\left(3 x \right)}}{\cos^{\frac{5}{2}}{\left(3 x \right)}} - \frac{12 \left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) \sin{\left(3 x \right)}}{\cos^{\frac{3}{2}}{\left(3 x \right)}} - \frac{38 \left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right)}{\sqrt{\cos{\left(3 x \right)}}} + \frac{8 \left(9 \sin{\left(3 x \right)} + 13 \cos{\left(3 x \right)}\right)}{\sqrt{\cos{\left(3 x \right)}}}\right)}{8} - x \sqrt{\cos{\left(3 x \right)}} + \frac{3 \left(\frac{3 \sin{\left(3 x \right)}}{\sqrt{\cos{\left(3 x \right)}}} + \sqrt{\cos{\left(3 x \right)}}\right) \left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right)}{4 \cos{\left(3 x \right)}} + \frac{3 \left(3 \sin{\left(3 x \right)} - 4 \cos{\left(3 x \right)}\right)}{\sqrt{\cos{\left(3 x \right)}}} - \frac{9 \left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) \sin{\left(3 x \right)}}{2 \cos^{\frac{3}{2}}{\left(3 x \right)}} + \frac{3 \left(3 \sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right)}{2 \sqrt{\cos{\left(3 x \right)}}} + 3 \sqrt{\cos{\left(3 x \right)}}\right) e^{- \frac{3 x}{2}}

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Gráfico

Derivada de x*sqrt(exp(-x)*cos(3x))exp(-x)

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Derivada de xsqrt(exp(-x)cos(3x))exp(-x)

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