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x^(sin^3)x
Derivada de la función/ sin^3/ x^(sin^3)x

Derivada de x^(sin^3)x

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

Ha introducido [src] 
    3     
 sin (x)  
x       *x
xxsin⁡3(x)x x^{\sin^{3}{\left(x \right)}} 
x^(sin(x)^3)*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)=xsin⁡3(x)f{\left(x \right)} = x^{\sin^{3}{\left(x \right)}}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. No logro encontrar los pasos en la búsqueda de esta derivada.

      Perola derivada

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

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

    1. Según el principio, aplicamos: xx tenemos 11

    Como resultado de: x(log⁡(sin⁡3(x))+1)(sin⁡3(x))sin⁡3(x)+xsin⁡3(x)x \left(\log{\left(\sin^{3}{\left(x \right)} \right)} + 1\right) \left(\sin^{3}{\left(x \right)}\right)^{\sin^{3}{\left(x \right)}} + x^{\sin^{3}{\left(x \right)}}

Respuesta:

x(log⁡(sin⁡3(x))+1)(sin⁡3(x))sin⁡3(x)+xsin⁡3(x)x \left(\log{\left(\sin^{3}{\left(x \right)} \right)} + 1\right) \left(\sin^{3}{\left(x \right)}\right)^{\sin^{3}{\left(x \right)}} + x^{\sin^{3}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-200200
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Primera derivada [src] 
    3            3    /   3                             \
 sin (x)      sin (x) |sin (x)        2                 |
x        + x*x       *|------- + 3*sin (x)*cos(x)*log(x)|
                      \   x                             /
xxsin⁡3(x)(3log⁡(x)sin⁡2(x)cos⁡(x)+sin⁡3(x)x)+xsin⁡3(x)x x^{\sin^{3}{\left(x \right)}} \left(3 \log{\left(x \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \frac{\sin^{3}{\left(x \right)}}{x}\right) + x^{\sin^{3}{\left(x \right)}}
Simplificar
Segunda derivada [src] 
    3    /  /                          2              2                                                           \                                      \       
 sin (x) |  |/sin(x)                  \     3      sin (x)        2                  2             6*cos(x)*sin(x)|     /sin(x)                  \       |       
x       *|x*||------ + 3*cos(x)*log(x)| *sin (x) - ------- - 3*sin (x)*log(x) + 6*cos (x)*log(x) + ---------------| + 2*|------ + 3*cos(x)*log(x)|*sin(x)|*sin(x)
         |  |\  x                     /                2                                                  x       |     \  x                     /       |       
         \  \                                         x                                                           /                                      /
xsin⁡3(x)(x((3log⁡(x)cos⁡(x)+sin⁡(x)x)2sin⁡3(x)−3log⁡(x)sin⁡2(x)+6log⁡(x)cos⁡2(x)+6sin⁡(x)cos⁡(x)x−sin⁡2(x)x2)+2(3log⁡(x)cos⁡(x)+sin⁡(x)x)sin⁡(x))sin⁡(x)x^{\sin^{3}{\left(x \right)}} \left(x \left(\left(3 \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{2} \sin^{3}{\left(x \right)} - 3 \log{\left(x \right)} \sin^{2}{\left(x \right)} + 6 \log{\left(x \right)} \cos^{2}{\left(x \right)} + \frac{6 \sin{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) + 2 \left(3 \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \sin{\left(x \right)}\right) \sin{\left(x \right)}
Simplificar
Tercera derivada [src] 
    3    /  /                          3                3           3                                                         2                                                  /   2                                                           \         2          \     /                          2              2                                                           \       \
 sin (x) |  |/sin(x)                  \     6      9*sin (x)   2*sin (x)        3                   2                    9*sin (x)*cos(x)        3    /sin(x)                  \ |sin (x)        2                  2             6*cos(x)*sin(x)|   18*cos (x)*sin(x)|     |/sin(x)                  \     3      sin (x)        2                  2             6*cos(x)*sin(x)|       |
x       *|x*||------ + 3*cos(x)*log(x)| *sin (x) - --------- + --------- + 6*cos (x)*log(x) - 21*sin (x)*cos(x)*log(x) - ---------------- - 3*sin (x)*|------ + 3*cos(x)*log(x)|*|------- - 6*cos (x)*log(x) + 3*sin (x)*log(x) - ---------------| + -----------------| + 3*||------ + 3*cos(x)*log(x)| *sin (x) - ------- - 3*sin (x)*log(x) + 6*cos (x)*log(x) + ---------------|*sin(x)|
         |  |\  x                     /                x            3                                                            2                    \  x                     / |    2                                                  x       |           x        |     |\  x                     /                2                                                  x       |       |
         \  \                                                      x                                                            x                                                \   x                                                           /                    /     \                                         x                                                           /       /
xsin⁡3(x)(x((3log⁡(x)cos⁡(x)+sin⁡(x)x)3sin⁡6(x)−3(3log⁡(x)cos⁡(x)+sin⁡(x)x)(3log⁡(x)sin⁡2(x)−6log⁡(x)cos⁡2(x)−6sin⁡(x)cos⁡(x)x+sin⁡2(x)x2)sin⁡3(x)−21log⁡(x)sin⁡2(x)cos⁡(x)+6log⁡(x)cos⁡3(x)−9sin⁡3(x)x+18sin⁡(x)cos⁡2(x)x−9sin⁡2(x)cos⁡(x)x2+2sin⁡3(x)x3)+3((3log⁡(x)cos⁡(x)+sin⁡(x)x)2sin⁡3(x)−3log⁡(x)sin⁡2(x)+6log⁡(x)cos⁡2(x)+6sin⁡(x)cos⁡(x)x−sin⁡2(x)x2)sin⁡(x))x^{\sin^{3}{\left(x \right)}} \left(x \left(\left(3 \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{3} \sin^{6}{\left(x \right)} - 3 \left(3 \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \left(3 \log{\left(x \right)} \sin^{2}{\left(x \right)} - 6 \log{\left(x \right)} \cos^{2}{\left(x \right)} - \frac{6 \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) \sin^{3}{\left(x \right)} - 21 \log{\left(x \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 6 \log{\left(x \right)} \cos^{3}{\left(x \right)} - \frac{9 \sin^{3}{\left(x \right)}}{x} + \frac{18 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{x} - \frac{9 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{2 \sin^{3}{\left(x \right)}}{x^{3}}\right) + 3 \left(\left(3 \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{2} \sin^{3}{\left(x \right)} - 3 \log{\left(x \right)} \sin^{2}{\left(x \right)} + 6 \log{\left(x \right)} \cos^{2}{\left(x \right)} + \frac{6 \sin{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) \sin{\left(x \right)}\right)
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Gráfico
Derivada de x^(sin^3)x
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