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Derivada de:
Derivada de (-4)/x^2
Derivada de 2/x²
Derivada de -2*y
Derivada de (3+2x)/(x-5)
Expresiones idénticas
y= tres ^((sin^ cuatro)(xlnx))
y es igual a 3 en el grado (( seno de en el grado 4)(xlnx))
y es igual a tres en el grado (( seno de en el grado cuatro)(xlnx))
y=3((sin4)(xlnx))
y=3sin4xlnx
y=3^((sin⁴)(xlnx))
y=3^sin^4xlnx
Expresiones con funciones
Seno sin
sin(x)^4
sin(x/3+3)+2*x
sin(x)^sin(x)
sin(x)/(1-x)
sin(x^2+3)
xlnx
xlnx-c
xlnx+arcsin(x^(1/2))
xlnx.
xlnx+√(x^2−1)
xlnx-3x

Derivada de la función/ sin^4/ (xlnx)/ y=3^((sin^4)(xlnx))

Derivada de y=3^((sin^4)(xlnx))

Solución

Ha introducido [src]

    4            
 sin (x)*x*log(x)
3

$$3^{x \log{\left(x \right)} \sin^{4}{\left(x \right)}}$$

3^(sin(x)^4*(x*log(x)))

Solución detallada

Sustituimos $u = x \log{\left(x \right)} \sin^{4}{\left(x \right)}$ .
$\frac{d}{d u} 3^{u} = 3^{u} \log{\left(3 \right)}$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} x \log{\left(x \right)} \sin^{4}{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \sin^{4}{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Sustituimos $u = \sin{\left(x \right)}$ .
  2. Según el principio, aplicamos: $u^{4}$ tenemos $4 u^{3}$
  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:
    
    $4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}$
  $g{\left(x \right)} = x \log{\left(x \right)}$ ; 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 $1$
    $g{\left(x \right)} = \log{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
    1. Derivado $\log{\left(x \right)}$ es $\frac{1}{x}$ .
    Como resultado de: $\log{\left(x \right)} + 1$
  Como resultado de: $4 x \log{\left(x \right)} \sin^{3}{\left(x \right)} \cos{\left(x \right)} + \left(\log{\left(x \right)} + 1\right) \sin^{4}{\left(x \right)}$
Como resultado de la secuencia de reglas:

$3^{x \log{\left(x \right)} \sin^{4}{\left(x \right)}} \left(4 x \log{\left(x \right)} \sin^{3}{\left(x \right)} \cos{\left(x \right)} + \left(\log{\left(x \right)} + 1\right) \sin^{4}{\left(x \right)}\right) \log{\left(3 \right)}$
Simplificamos:

$3^{x \log{\left(x \right)} \sin^{4}{\left(x \right)}} \left(4 x \log{\left(x \right)} \cos{\left(x \right)} + \left(\log{\left(x \right)} + 1\right) \sin{\left(x \right)}\right) \log{\left(3 \right)} \sin^{3}{\left(x \right)}$

Respuesta:

$3^{x \log{\left(x \right)} \sin^{4}{\left(x \right)}} \left(4 x \log{\left(x \right)} \cos{\left(x \right)} + \left(\log{\left(x \right)} + 1\right) \sin{\left(x \right)}\right) \log{\left(3 \right)} \sin^{3}{\left(x \right)}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

    4                                                                      
 sin (x)*x*log(x) /   4                          3                 \       
3                *\sin (x)*(1 + log(x)) + 4*x*sin (x)*cos(x)*log(x)/*log(3)

$$3^{x \log{\left(x \right)} \sin^{4}{\left(x \right)}} \left(4 x \log{\left(x \right)} \sin^{3}{\left(x \right)} \cos{\left(x \right)} + \left(\log{\left(x \right)} + 1\right) \sin^{4}{\left(x \right)}\right) \log{\left(3 \right)}$$

Simplificar

Segunda derivada [src]

      4                   /   2                                                                                                                                                                                    \       
 x*sin (x)*log(x)    2    |sin (x)                                                              2    4                    2                                                                             2          |       
3                *sin (x)*|------- + 4*cos(x)*sin(x) + ((1 + log(x))*sin(x) + 4*x*cos(x)*log(x)) *sin (x)*log(3) - 4*x*sin (x)*log(x) + 4*(1 + log(x))*cos(x)*sin(x) + 4*cos(x)*log(x)*sin(x) + 12*x*cos (x)*log(x)|*log(3)
                          \   x                                                                                                                                                                                    /

$$3^{x \log{\left(x \right)} \sin^{4}{\left(x \right)}} \left(- 4 x \log{\left(x \right)} \sin^{2}{\left(x \right)} + 12 x \log{\left(x \right)} \cos^{2}{\left(x \right)} + \left(4 x \log{\left(x \right)} \cos{\left(x \right)} + \left(\log{\left(x \right)} + 1\right) \sin{\left(x \right)}\right)^{2} \log{\left(3 \right)} \sin^{4}{\left(x \right)} + 4 \left(\log{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} + 4 \log{\left(x \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\sin^{2}{\left(x \right)}}{x}\right) \log{\left(3 \right)} \sin^{2}{\left(x \right)}$$

Simplificar

Tercera derivada [src]

      4           /                 3                                                                                                                                         2                                                                                                                                                                                /   2                                                                                                                        \       \              
 x*sin (x)*log(x) |       3      sin (x)        3                  3                         2                                                      3    2       8      12*sin (x)*cos(x)         2                                  3                   2                            2                         4                                              |sin (x)                            2                                                                             2          |       |              
3                *|- 8*sin (x) - ------- - 8*sin (x)*log(x) - 4*sin (x)*(1 + log(x)) + 24*cos (x)*sin(x) + ((1 + log(x))*sin(x) + 4*x*cos(x)*log(x)) *log (3)*sin (x) + ----------------- + 12*cos (x)*(1 + log(x))*sin(x) + 24*x*cos (x)*log(x) + 24*cos (x)*log(x)*sin(x) - 40*x*sin (x)*cos(x)*log(x) + 3*sin (x)*((1 + log(x))*sin(x) + 4*x*cos(x)*log(x))*|------- + 4*cos(x)*sin(x) - 4*x*sin (x)*log(x) + 4*(1 + log(x))*cos(x)*sin(x) + 4*cos(x)*log(x)*sin(x) + 12*x*cos (x)*log(x)|*log(3)|*log(3)*sin(x)
                  |                  2                                                                                                                                          x                                                                                                                                                                              \   x                                                                                                                        /       |              
                  \                 x                                                                                                                                                                                                                                                                                                                                                                                                                                                               /

$$3^{x \log{\left(x \right)} \sin^{4}{\left(x \right)}} \left(- 40 x \log{\left(x \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 24 x \log{\left(x \right)} \cos^{3}{\left(x \right)} + \left(4 x \log{\left(x \right)} \cos{\left(x \right)} + \left(\log{\left(x \right)} + 1\right) \sin{\left(x \right)}\right)^{3} \log{\left(3 \right)}^{2} \sin^{8}{\left(x \right)} + 3 \left(4 x \log{\left(x \right)} \cos{\left(x \right)} + \left(\log{\left(x \right)} + 1\right) \sin{\left(x \right)}\right) \left(- 4 x \log{\left(x \right)} \sin^{2}{\left(x \right)} + 12 x \log{\left(x \right)} \cos^{2}{\left(x \right)} + 4 \left(\log{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} + 4 \log{\left(x \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\sin^{2}{\left(x \right)}}{x}\right) \log{\left(3 \right)} \sin^{4}{\left(x \right)} - 4 \left(\log{\left(x \right)} + 1\right) \sin^{3}{\left(x \right)} + 12 \left(\log{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos^{2}{\left(x \right)} - 8 \log{\left(x \right)} \sin^{3}{\left(x \right)} + 24 \log{\left(x \right)} \sin{\left(x \right)} \cos^{2}{\left(x \right)} - 8 \sin^{3}{\left(x \right)} + 24 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \frac{12 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{\sin^{3}{\left(x \right)}}{x^{2}}\right) \log{\left(3 \right)} \sin{\left(x \right)}$$

Simplificar

Gráfico

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Expresiones idénticas

Expresiones con funciones

Derivada de y=3^((sin^4)(xlnx))

Gráfico:

Definida a trozos:

Solución