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

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

Función f() - derivada -er orden en el punto
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Solución

Ha introducido [src] 
    4            
 sin (x)         
3       *x*log(x)
3sin⁡4(x)xlog⁡(x)3^{\sin^{4}{\left(x \right)}} x \log{\left(x \right)} 
3^(sin(x)^4)*(x*log(x))
Solución detallada

  1. Se aplica la regla de la derivada de una multiplicación:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=3sin⁡4(x)f{\left(x \right)} = 3^{\sin^{4}{\left(x \right)}}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Sustituimos u=sin⁡4(x)u = \sin^{4}{\left(x \right)}.

    2. ddu3u=3ulog⁡(3)\frac{d}{d u} 3^{u} = 3^{u} \log{\left(3 \right)}

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddxsin⁡4(x)\frac{d}{d x} \sin^{4}{\left(x \right)}:

      1. Sustituimos u=sin⁡(x)u = \sin{\left(x \right)}.

      2. Según el principio, aplicamos: u4u^{4} tenemos 4u34 u^{3}

      3. Luego se aplica una cadena de reglas. Multiplicamos por ddxsin⁡(x)\frac{d}{d x} \sin{\left(x \right)}:

        1. La derivada del seno es igual al coseno:

          ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

        Como resultado de la secuencia de reglas:

        4sin⁡3(x)cos⁡(x)4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}

      Como resultado de la secuencia de reglas:

      4⋅3sin⁡4(x)log⁡(3)sin⁡3(x)cos⁡(x)4 \cdot 3^{\sin^{4}{\left(x \right)}} \log{\left(3 \right)} \sin^{3}{\left(x \right)} \cos{\left(x \right)}

    g(x)=xlog⁡(x)g{\left(x \right)} = x \log{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Se aplica la regla de la derivada de una multiplicación:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=xf{\left(x \right)} = x; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Según el principio, aplicamos: xx tenemos 11

      g(x)=log⁡(x)g{\left(x \right)} = \log{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Derivado log⁡(x)\log{\left(x \right)} es 1x\frac{1}{x}.

      Como resultado de: log⁡(x)+1\log{\left(x \right)} + 1

    Como resultado de: 4⋅3sin⁡4(x)xlog⁡(3)log⁡(x)sin⁡3(x)cos⁡(x)+3sin⁡4(x)(log⁡(x)+1)4 \cdot 3^{\sin^{4}{\left(x \right)}} x \log{\left(3 \right)} \log{\left(x \right)} \sin^{3}{\left(x \right)} \cos{\left(x \right)} + 3^{\sin^{4}{\left(x \right)}} \left(\log{\left(x \right)} + 1\right)

  2. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-100100
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Primera derivada [src] 
    4                           4                                
 sin (x)                     sin (x)    3                        
3       *(1 + log(x)) + 4*x*3       *sin (x)*cos(x)*log(3)*log(x)
4⋅3sin⁡4(x)xlog⁡(3)log⁡(x)sin⁡3(x)cos⁡(x)+3sin⁡4(x)(log⁡(x)+1)4 \cdot 3^{\sin^{4}{\left(x \right)}} x \log{\left(3 \right)} \log{\left(x \right)} \sin^{3}{\left(x \right)} \cos{\left(x \right)} + 3^{\sin^{4}{\left(x \right)}} \left(\log{\left(x \right)} + 1\right)
Simplificar
Segunda derivada [src] 
    4                                                                                                                             
 sin (x) /1        3                                        2    /     2           2           2       4          \              \
3       *|- + 8*sin (x)*(1 + log(x))*cos(x)*log(3) + 4*x*sin (x)*\- sin (x) + 3*cos (x) + 4*cos (x)*sin (x)*log(3)/*log(3)*log(x)|
         \x                                                                                                                      /
3sin⁡4(x)(4x(4log⁡(3)sin⁡4(x)cos⁡2(x)−sin⁡2(x)+3cos⁡2(x))log⁡(3)log⁡(x)sin⁡2(x)+8(log⁡(x)+1)log⁡(3)sin⁡3(x)cos⁡(x)+1x)3^{\sin^{4}{\left(x \right)}} \left(4 x \left(4 \log{\left(3 \right)} \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)} - \sin^{2}{\left(x \right)} + 3 \cos^{2}{\left(x \right)}\right) \log{\left(3 \right)} \log{\left(x \right)} \sin^{2}{\left(x \right)} + 8 \left(\log{\left(x \right)} + 1\right) \log{\left(3 \right)} \sin^{3}{\left(x \right)} \cos{\left(x \right)} + \frac{1}{x}\right)
Simplificar
Tercera derivada [src] 
    4    /             3                                                                                                                                                                                                                                            \
 sin (x) |  1    12*sin (x)*cos(x)*log(3)         2                 /     2           2           2       4          \              /       2           2           6                  2       2       8            2       4          \                            |
3       *|- -- + ------------------------ + 12*sin (x)*(1 + log(x))*\- sin (x) + 3*cos (x) + 4*cos (x)*sin (x)*log(3)/*log(3) + 8*x*\- 5*sin (x) + 3*cos (x) - 6*sin (x)*log(3) + 8*cos (x)*log (3)*sin (x) + 18*cos (x)*sin (x)*log(3)/*cos(x)*log(3)*log(x)*sin(x)|
         |   2              x                                                                                                                                                                                                                                       |
         \  x                                                                                                                                                                                                                                                       /
3sin⁡4(x)(8x(8log⁡(3)2sin⁡8(x)cos⁡2(x)−6log⁡(3)sin⁡6(x)+18log⁡(3)sin⁡4(x)cos⁡2(x)−5sin⁡2(x)+3cos⁡2(x))log⁡(3)log⁡(x)sin⁡(x)cos⁡(x)+12(log⁡(x)+1)(4log⁡(3)sin⁡4(x)cos⁡2(x)−sin⁡2(x)+3cos⁡2(x))log⁡(3)sin⁡2(x)+12log⁡(3)sin⁡3(x)cos⁡(x)x−1x2)3^{\sin^{4}{\left(x \right)}} \left(8 x \left(8 \log{\left(3 \right)}^{2} \sin^{8}{\left(x \right)} \cos^{2}{\left(x \right)} - 6 \log{\left(3 \right)} \sin^{6}{\left(x \right)} + 18 \log{\left(3 \right)} \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)} - 5 \sin^{2}{\left(x \right)} + 3 \cos^{2}{\left(x \right)}\right) \log{\left(3 \right)} \log{\left(x \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 12 \left(\log{\left(x \right)} + 1\right) \left(4 \log{\left(3 \right)} \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)} - \sin^{2}{\left(x \right)} + 3 \cos^{2}{\left(x \right)}\right) \log{\left(3 \right)} \sin^{2}{\left(x \right)} + \frac{12 \log{\left(3 \right)} \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{1}{x^{2}}\right)
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Gráfico
Derivada de y=3^(sin^4)(xlnx)
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