Derivada de xsinx^8lnx(sinx)

Solución

Ha introducido [src]

     8                 
x*sin (x)*log(x)*sin(x)

$$x \sin^{8}{\left(x \right)} \log{\left(x \right)} \sin{\left(x \right)}$$

((x*sin(x)^8)*log(x))*sin(x)

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = x \sin^{8}{\left(x \right)} \log{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x \sin^{8}{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  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 $1$
    $g{\left(x \right)} = \sin^{8}{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
    1. Sustituimos $u = \sin{\left(x \right)}$ .
    2. Según el principio, aplicamos: $u^{8}$ tenemos $8 u^{7}$
    3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \sin{\left(x \right)}$ :
      1. La derivada del seno es igual al coseno:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      Como resultado de la secuencia de reglas:
      
      $8 \sin^{7}{\left(x \right)} \cos{\left(x \right)}$
    Como resultado de: $8 x \sin^{7}{\left(x \right)} \cos{\left(x \right)} + \sin^{8}{\left(x \right)}$
  $g{\left(x \right)} = \log{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Derivado $\log{\left(x \right)}$ es $\frac{1}{x}$ .
  Como resultado de: $\left(8 x \sin^{7}{\left(x \right)} \cos{\left(x \right)} + \sin^{8}{\left(x \right)}\right) \log{\left(x \right)} + \sin^{8}{\left(x \right)}$
$g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $x \log{\left(x \right)} \sin^{8}{\left(x \right)} \cos{\left(x \right)} + \left(\left(8 x \sin^{7}{\left(x \right)} \cos{\left(x \right)} + \sin^{8}{\left(x \right)}\right) \log{\left(x \right)} + \sin^{8}{\left(x \right)}\right) \sin{\left(x \right)}$
Simplificamos:

$\left(9 x \log{\left(x \right)} \cos{\left(x \right)} + \log{\left(x \right)} \sin{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{8}{\left(x \right)}$

Respuesta:

$\left(9 x \log{\left(x \right)} \cos{\left(x \right)} + \log{\left(x \right)} \sin{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{8}{\left(x \right)}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

/   8      /   8             7          \       \               8                 
\sin (x) + \sin (x) + 8*x*sin (x)*cos(x)/*log(x)/*sin(x) + x*sin (x)*cos(x)*log(x)

$$x \log{\left(x \right)} \sin^{8}{\left(x \right)} \cos{\left(x \right)} + \left(\left(8 x \sin^{7}{\left(x \right)} \cos{\left(x \right)} + \sin^{8}{\left(x \right)}\right) \log{\left(x \right)} + \sin^{8}{\left(x \right)}\right) \sin{\left(x \right)}$$

Simplificar

Segunda derivada [src]

        /     2                                                                                                                                                                 \
   7    |  sin (x)     /  /   2           2   \                  \                                                                  2             2*(8*x*cos(x) + sin(x))*sin(x)|
sin (x)*|- ------- - 8*\x*\sin (x) - 7*cos (x)/ - 2*cos(x)*sin(x)/*log(x) + 2*((8*x*cos(x) + sin(x))*log(x) + sin(x))*cos(x) - x*sin (x)*log(x) + ------------------------------|
        \     x                                                                                                                                                 x               /

$$\left(- x \log{\left(x \right)} \sin^{2}{\left(x \right)} - 8 \left(x \left(\sin^{2}{\left(x \right)} - 7 \cos^{2}{\left(x \right)}\right) - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(x \right)} + 2 \left(\left(8 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \right)} + \frac{2 \left(8 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{x} - \frac{\sin^{2}{\left(x \right)}}{x}\right) \sin^{7}{\left(x \right)}$$

Simplificar

Tercera derivada [src]

         /       3                                                            /   2                                                                                           \                                                                                                    2                               /  /   2           2   \                  \                                 \
    6    |  2*sin (x)        2                                                |sin (x)     /  /   2           2   \                  \          2*(8*x*cos(x) + sin(x))*sin(x)|            /  /   2           2   \              /        2            2   \       \          3*sin (x)*(8*x*cos(x) + sin(x))   24*\x*\sin (x) - 7*cos (x)/ - 2*cos(x)*sin(x)/*sin(x)        2                 |
-sin (x)*|- --------- + 3*sin (x)*((8*x*cos(x) + sin(x))*log(x) + sin(x)) + 3*|------- + 8*\x*\sin (x) - 7*cos (x)/ - 2*cos(x)*sin(x)/*log(x) - ------------------------------|*cos(x) + 8*\3*\sin (x) - 7*cos (x)/*sin(x) + 2*x*\- 21*cos (x) + 11*sin (x)/*cos(x)/*log(x) + ------------------------------- + ----------------------------------------------------- + x*sin (x)*cos(x)*log(x)|
         |       2                                                            \   x                                                                           x               /                                                                                                               2                                           x                                                    |
         \      x                                                                                                                                                                                                                                                                            x                                                                                                 /

$$- \left(x \log{\left(x \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 3 \left(\left(8 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{2}{\left(x \right)} + 8 \left(2 x \left(11 \sin^{2}{\left(x \right)} - 21 \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)} + 3 \left(\sin^{2}{\left(x \right)} - 7 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}\right) \log{\left(x \right)} + 3 \left(8 \left(x \left(\sin^{2}{\left(x \right)} - 7 \cos^{2}{\left(x \right)}\right) - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(x \right)} - \frac{2 \left(8 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{x} + \frac{\sin^{2}{\left(x \right)}}{x}\right) \cos{\left(x \right)} + \frac{24 \left(x \left(\sin^{2}{\left(x \right)} - 7 \cos^{2}{\left(x \right)}\right) - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \sin{\left(x \right)}}{x} + \frac{3 \left(8 x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{x^{2}} - \frac{2 \sin^{3}{\left(x \right)}}{x^{2}}\right) \sin^{6}{\left(x \right)}$$

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Derivada de xsinx^8lnx(sinx)

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