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Derivada de 5^10
Derivada de 2*cos(3*x)
Derivada de x^(7/6)
Derivada de x^(2/5)
Expresiones idénticas
y=(x*sin(x))^8lnsin(x)
y es igual a (x multiplicar por seno de (x)) en el grado 8ln seno de (x)
y=(x*sin(x))8lnsin(x)
y=x*sinx8lnsinx
y=(x*sin(x))⁸lnsin(x)
y=(xsin(x))^8lnsin(x)
y=(xsin(x))8lnsin(x)
y=xsinx8lnsinx
y=xsinx^8lnsinx
Expresiones semejantes
y=(x*sinx)^8lnsinx
Expresiones con funciones
Seno sin
sin(t)*cos(t)
sin(x)^(23)
sin(5*x+2)
sin(2*y)
sin(x)^(cos(x))

Derivada de la función/ x*sin/ sin(x)/ y=(x*sin(x))^8lnsin(x)

Derivada de y=(x*sin(x))^8lnsin(x)

Solución

Ha introducido [src]

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

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

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

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

 8    8                                                           
x *sin (x)*cos(x)    7    7                                       
----------------- + x *sin (x)*(8*sin(x) + 8*x*cos(x))*log(sin(x))
      sin(x)

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

Simplificar

Segunda derivada [src]

           /                                                                                                                                                      /       2   \                                  \
 6    6    |    /                       2                                                                                              \                2    2    |    cos (x)|                                  |
x *sin (x)*|- 8*\- 8*(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)/*log(sin(x)) - x *sin (x)*|1 + -------| + 16*x*(x*cos(x) + sin(x))*cos(x)|
           |                                                                                                                                                      |       2   |                                  |
           \                                                                                                                                                      \    sin (x)/                                  /

x^{6} \left(- x^{2} \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin^{2}{\left(x \right)} + 16 x \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \right)} - 8 \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)} - 8 \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) \log{\left(\sin{\left(x \right)} \right)}\right) \sin^{6}{\left(x \right)}

Simplificar

Tercera derivada [src]

             /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      /       2   \                        /       2   \                    \
   5    5    |    /                        /     2       2    2         2    2                        \        2                                               2           2    2                             2    2                                 2                                2    2                                                 2             2                                                                                                                                      \                    /                       2                                                                                              \           3    2    |    cos (x)|              2    2    |    cos (x)|                    |
2*x *sin (x)*|- 4*\- 8*(x*cos(x) + sin(x))*\7*sin (x) - x *sin (x) + 7*x *cos (x) + 16*x*cos(x)*sin(x)/ + 6*sin (x)*(x*cos(x) + sin(x)) + 8*(x*cos(x) + sin(x)) *sin(x) + x *sin (x)*(3*sin(x) + x*cos(x)) - x *sin (x)*(x*cos(x) + sin(x)) + 6*x*sin (x)*(-2*cos(x) + x*sin(x)) + 6*x *cos (x)*(x*cos(x) + sin(x)) + 8*x*(x*cos(x) + sin(x)) *cos(x) + 6*x *(-2*cos(x) + x*sin(x))*cos(x)*sin(x) + 8*x*(-2*cos(x) + x*sin(x))*(x*cos(x) + sin(x))*sin(x) + 14*x*(x*cos(x) + sin(x))*cos(x)*sin(x)/*log(sin(x)) - 12*x*\- 8*(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)/*cos(x) + x *sin (x)*|1 + -------|*cos(x) - 12*x *sin (x)*|1 + -------|*(x*cos(x) + sin(x))|
             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |       2   |                        |       2   |                    |
             \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      \    sin (x)/                        \    sin (x)/                    /

2 x^{5} \left(x^{3} \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 12 x^{2} \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{2}{\left(x \right)} - 12 x \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)} - 8 \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) \cos{\left(x \right)} - 4 \left(6 x^{2} \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)} - x^{2} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{2}{\left(x \right)} + 6 x^{2} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos^{2}{\left(x \right)} + x^{2} \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \sin^{2}{\left(x \right)} + 8 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)} + 6 x \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \sin^{2}{\left(x \right)} + 8 x \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \cos{\left(x \right)} + 14 x \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)} + 8 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \sin{\left(x \right)} - 8 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(- x^{2} \sin^{2}{\left(x \right)} + 7 x^{2} \cos^{2}{\left(x \right)} + 16 x \sin{\left(x \right)} \cos{\left(x \right)} + 7 \sin^{2}{\left(x \right)}\right) + 6 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{2}{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}\right) \sin^{5}{\left(x \right)}

Simplificar

Gráfico

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Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

Derivada de y=(x*sin(x))^8lnsin(x)

Gráfico:

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Solución