Derivada de y=(xsinx)^81n(xsinx)

Solución

Ha introducido [src]

          81           
(x*sin(x))  *n*x*sin(x)

$$n \left(x \sin{\left(x \right)}\right)^{81} x \sin{\left(x \right)}$$

((x*sin(x))^81*n)*(x*sin(x))

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = n \left(x \sin{\left(x \right)}\right)^{81}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
  1. Sustituimos $u = x \sin{\left(x \right)}$ .
  2. Según el principio, aplicamos: $u^{81}$ tenemos $81 u^{80}$
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} x \sin{\left(x \right)}$ :
    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 $1$
      $g{\left(x \right)} = \sin{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
      1. La derivada del seno es igual al coseno:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      Como resultado de: $x \cos{\left(x \right)} + \sin{\left(x \right)}$
    Como resultado de la secuencia de reglas:
    
    $81 x^{80} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{80}{\left(x \right)}$
  Entonces, como resultado: $81 n x^{80} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{80}{\left(x \right)}$
$g{\left(x \right)} = x \sin{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
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 $1$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. La derivada del seno es igual al coseno:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  Como resultado de: $x \cos{\left(x \right)} + \sin{\left(x \right)}$
Como resultado de: $81 n x^{81} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{81}{\left(x \right)} + n x^{81} \sin^{81}{\left(x \right)} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)$
Simplificamos:

$82 n x^{81} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{81}{\left(x \right)}$

Respuesta:

$82 n x^{81} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{81}{\left(x \right)}$

Primera derivada [src]

   81    81                             81    81                             
n*x  *sin  (x)*(x*cos(x) + sin(x)) + n*x  *sin  (x)*(81*sin(x) + 81*x*cos(x))

$$n x^{81} \left(81 x \cos{\left(x \right)} + 81 \sin{\left(x \right)}\right) \sin^{81}{\left(x \right)} + n x^{81} \sin^{81}{\left(x \right)} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)$$

Simplificar

Segunda derivada [src]

   80    80    /                        2                                                                                                       \
n*x  *sin  (x)*\6723*(x*cos(x) + sin(x))  - 81*(x*cos(x) + sin(x))*sin(x) - 82*x*(-2*cos(x) + x*sin(x))*sin(x) - 81*x*(x*cos(x) + sin(x))*cos(x)/

$$n x^{80} \left(- 82 x \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin{\left(x \right)} - 81 x \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \right)} + 6723 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} - 81 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}\right) \sin^{80}{\left(x \right)}$$

Simplificar

Tercera derivada [src]

    79    79    /                           /      2       2    2          2    2                         \                           /                        2                                                                                              \           2                                                  2              2    2                              2    2                                      2                                   2    2                                                    2                2                                                                                                                                            \
-n*x  *sin  (x)*\- 6561*(x*cos(x) + sin(x))*\80*sin (x) - x *sin (x) + 80*x *cos (x) + 162*x*cos(x)*sin(x)/ + 243*(x*cos(x) + sin(x))*\- 81*(x*cos(x) + sin(x))  + (x*cos(x) + sin(x))*sin(x) + x*(-2*cos(x) + x*sin(x))*sin(x) + x*(x*cos(x) + sin(x))*cos(x)/ + 6399*sin (x)*(x*cos(x) + sin(x)) + 6561*(x*cos(x) + sin(x)) *sin(x) - 81*x *sin (x)*(x*cos(x) + sin(x)) + 82*x *sin (x)*(3*sin(x) + x*cos(x)) + 6399*x*sin (x)*(-2*cos(x) + x*sin(x)) + 6399*x *cos (x)*(x*cos(x) + sin(x)) + 6561*x*(x*cos(x) + sin(x)) *cos(x) + 6399*x *(-2*cos(x) + x*sin(x))*cos(x)*sin(x) + 6804*x*(-2*cos(x) + x*sin(x))*(x*cos(x) + sin(x))*sin(x) + 12960*x*(x*cos(x) + sin(x))*cos(x)*sin(x)/

$$- n x^{79} \left(6399 x^{2} \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)} - 81 x^{2} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{2}{\left(x \right)} + 6399 x^{2} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos^{2}{\left(x \right)} + 82 x^{2} \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \sin^{2}{\left(x \right)} + 6804 x \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)} + 6399 x \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin^{2}{\left(x \right)} + 6561 x \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \cos{\left(x \right)} + 12960 x \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)} + 6561 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \sin{\left(x \right)} - 6561 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(- x^{2} \sin^{2}{\left(x \right)} + 80 x^{2} \cos^{2}{\left(x \right)} + 162 x \sin{\left(x \right)} \cos{\left(x \right)} + 80 \sin^{2}{\left(x \right)}\right) + 243 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(x \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin{\left(x \right)} + x \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \right)} - 81 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}\right) + 6399 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{2}{\left(x \right)}\right) \sin^{79}{\left(x \right)}$$

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Derivada de y=(xsinx)^81n(xsinx)

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