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Derivada de la función/ sin(x+(cos(x))^(2/3))/ x*sin(x+(cos(x))^(2/3))exp(-x)

Derivada de x*sin(x+(cos(x))^(2/3))exp(-x)

Solución

Ha introducido [src]

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

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

(x*sin(x + cos(x)^(2/3)))*exp(-x)

Solución detallada

Se aplica la regla de la derivada parcial:

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

$f{\left(x \right)} = x \sin{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)}$ 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:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Según el principio, aplicamos: $x$ tenemos $1$
  $g{\left(x \right)} = \sin{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Sustituimos $u = x + \cos^{\frac{2}{3}}{\left(x \right)}$ .
  2. La derivada del seno es igual al coseno:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \left(x + \cos^{\frac{2}{3}}{\left(x \right)}\right)$ :
    1. diferenciamos $x + \cos^{\frac{2}{3}}{\left(x \right)}$ miembro por miembro:
      1. Según el principio, aplicamos: $x$ tenemos $1$
      2. Sustituimos $u = \cos{\left(x \right)}$ .
      3. Según el principio, aplicamos: $u^{\frac{2}{3}}$ tenemos $\frac{2}{3 \sqrt[3]{u}}$
      4. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \cos{\left(x \right)}$ :
        
        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 la secuencia de reglas:
        
        $- \frac{2 \sin{\left(x \right)}}{3 \sqrt[3]{\cos{\left(x \right)}}}$
      Como resultado de: $- \frac{2 \sin{\left(x \right)}}{3 \sqrt[3]{\cos{\left(x \right)}}} + 1$
    Como resultado de la secuencia de reglas:
    
    $\left(- \frac{2 \sin{\left(x \right)}}{3 \sqrt[3]{\cos{\left(x \right)}}} + 1\right) \cos{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)}$
  Como resultado de: $x \left(- \frac{2 \sin{\left(x \right)}}{3 \sqrt[3]{\cos{\left(x \right)}}} + 1\right) \cos{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)} + \sin{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)}$
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:

$\left(- x e^{x} \sin{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)} + \left(x \left(- \frac{2 \sin{\left(x \right)}}{3 \sqrt[3]{\cos{\left(x \right)}}} + 1\right) \cos{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)} + \sin{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)}\right) e^{x}\right) e^{- 2 x}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

$$- x e^{- x} \sin{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)} + \left(x \left(- \frac{2 \sin{\left(x \right)}}{3 \sqrt[3]{\cos{\left(x \right)}}} + 1\right) \cos{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)} + \sin{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)}\right) e^{- x}$$

Simplificar

Segunda derivada [src]

/                                                                                           /                 2                        /                  2    \                   \                                           \    
|                                                  /      2*sin(x) \    /       2/3   \     |/      2*sin(x) \     /       2/3   \     |     2/3       sin (x) |    /       2/3   \|       /      2*sin(x) \    /       2/3   \|    
|                                                2*|-3 + ----------|*cos\x + cos   (x)/   x*||-3 + ----------| *sin\x + cos   (x)/ + 2*|3*cos   (x) + ---------|*cos\x + cos   (x)/|   2*x*|-3 + ----------|*cos\x + cos   (x)/|    
|                                                  |     3 ________|                        ||     3 ________|                         |                 4/3   |                   |       |     3 ________|                   |    
|       /       2/3   \        /       2/3   \     \     \/ cos(x) /                        \\     \/ cos(x) /                         \              cos   (x)/                   /       \     \/ cos(x) /                   |  -x
|- 2*sin\x + cos   (x)/ + x*sin\x + cos   (x)/ - -------------------------------------- - ------------------------------------------------------------------------------------------ + ----------------------------------------|*e  
\                                                                  3                                                                  9                                                                   3                    /

$$\left(- \frac{x \left(\left(\frac{2 \sin{\left(x \right)}}{\sqrt[3]{\cos{\left(x \right)}}} - 3\right)^{2} \sin{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)} + 2 \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{\frac{4}{3}}{\left(x \right)}} + 3 \cos^{\frac{2}{3}}{\left(x \right)}\right) \cos{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)}\right)}{9} + \frac{2 x \left(\frac{2 \sin{\left(x \right)}}{\sqrt[3]{\cos{\left(x \right)}}} - 3\right) \cos{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)}}{3} + x \sin{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)} - \frac{2 \left(\frac{2 \sin{\left(x \right)}}{\sqrt[3]{\cos{\left(x \right)}}} - 3\right) \cos{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)}}{3} - 2 \sin{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)}\right) e^{- x}$$

Simplificar

Tercera derivada [src]

/                                                                                         /                  2    \                                       2                        /                   3                                          /                  2    \                           3       /       2/3   \\     /                 2                        /                  2    \                   \                                         \    
|                                                                                         |     2/3       sin (x) |    /       2/3   \   /      2*sin(x) \     /       2/3   \     |  /      2*sin(x) \     /       2/3   \     /      2*sin(x) \ |     2/3       sin (x) |    /       2/3   \   8*sin (x)*cos\x + cos   (x)/|     |/      2*sin(x) \     /       2/3   \     |     2/3       sin (x) |    /       2/3   \|                                         |    
|                                                                                       2*|3*cos   (x) + ---------|*cos\x + cos   (x)/   |-3 + ----------| *sin\x + cos   (x)/   x*|- |-3 + ----------| *cos\x + cos   (x)/ + 6*|-3 + ----------|*|3*cos   (x) + ---------|*sin\x + cos   (x)/ + ----------------------------|   x*||-3 + ----------| *sin\x + cos   (x)/ + 2*|3*cos   (x) + ---------|*cos\x + cos   (x)/|                                         |    
|                                                                                         |                 4/3   |                      |     3 ________|                         |  |     3 ________|                         |     3 ________| |                 4/3   |                                  7/3             |     ||     3 ________|                         |                 4/3   |                   |                                         |    
|     /       2/3   \        /       2/3   \     /      2*sin(x) \    /       2/3   \     \              cos   (x)/                      \     \/ cos(x) /                         \  \     \/ cos(x) /                         \     \/ cos(x) / \              cos   (x)/                               cos   (x)          /     \\     \/ cos(x) /                         \              cos   (x)/                   /     /      2*sin(x) \    /       2/3   \|  -x
|3*sin\x + cos   (x)/ - x*sin\x + cos   (x)/ + 2*|-3 + ----------|*cos\x + cos   (x)/ - ---------------------------------------------- - ------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------ - x*|-3 + ----------|*cos\x + cos   (x)/|*e  
|                                                |     3 ________|                                            3                                            3                                                                                           27                                                                                                                    3                                                  |     3 ________|                   |    
\                                                \     \/ cos(x) /                                                                                                                                                                                                                                                                                                                                                              \     \/ cos(x) /                   /

$$\left(\frac{x \left(\left(\frac{2 \sin{\left(x \right)}}{\sqrt[3]{\cos{\left(x \right)}}} - 3\right)^{2} \sin{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)} + 2 \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{\frac{4}{3}}{\left(x \right)}} + 3 \cos^{\frac{2}{3}}{\left(x \right)}\right) \cos{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)}\right)}{3} - x \left(\frac{2 \sin{\left(x \right)}}{\sqrt[3]{\cos{\left(x \right)}}} - 3\right) \cos{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)} - \frac{x \left(- \left(\frac{2 \sin{\left(x \right)}}{\sqrt[3]{\cos{\left(x \right)}}} - 3\right)^{3} \cos{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)} + 6 \left(\frac{2 \sin{\left(x \right)}}{\sqrt[3]{\cos{\left(x \right)}}} - 3\right) \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{\frac{4}{3}}{\left(x \right)}} + 3 \cos^{\frac{2}{3}}{\left(x \right)}\right) \sin{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)} + \frac{8 \sin^{3}{\left(x \right)} \cos{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)}}{\cos^{\frac{7}{3}}{\left(x \right)}}\right)}{27} - x \sin{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)} - \frac{\left(\frac{2 \sin{\left(x \right)}}{\sqrt[3]{\cos{\left(x \right)}}} - 3\right)^{2} \sin{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)}}{3} + 2 \left(\frac{2 \sin{\left(x \right)}}{\sqrt[3]{\cos{\left(x \right)}}} - 3\right) \cos{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)} - \frac{2 \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{\frac{4}{3}}{\left(x \right)}} + 3 \cos^{\frac{2}{3}}{\left(x \right)}\right) \cos{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)}}{3} + 3 \sin{\left(x + \cos^{\frac{2}{3}}{\left(x \right)} \right)}\right) e^{- x}$$

Simplificar

Gráfico

Derivada de x*sin(x+(cos(x))^(2/3))exp(-x)

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

Derivada de x*sin(x+(cos(x))^(2/3))exp(-x)

Gráfico:

Definida a trozos:

Solución