Derivada de x^(sin^3)x

Solución

Ha introducido [src]

    3     
 sin (x)  
x       *x

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

x^(sin(x)^3)*x

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = x^{\sin^{3}{\left(x \right)}}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $\left(\log{\left(\sin^{3}{\left(x \right)} \right)} + 1\right) \left(\sin^{3}{\left(x \right)}\right)^{\sin^{3}{\left(x \right)}}$
$g{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. Según el principio, aplicamos: $x$ tenemos $1$
Como resultado de: $x \left(\log{\left(\sin^{3}{\left(x \right)} \right)} + 1\right) \left(\sin^{3}{\left(x \right)}\right)^{\sin^{3}{\left(x \right)}} + x^{\sin^{3}{\left(x \right)}}$

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

$$x^{\sin^{3}{\left(x \right)}} \left(x \left(\left(3 \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{2} \sin^{3}{\left(x \right)} - 3 \log{\left(x \right)} \sin^{2}{\left(x \right)} + 6 \log{\left(x \right)} \cos^{2}{\left(x \right)} + \frac{6 \sin{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) + 2 \left(3 \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \sin{\left(x \right)}\right) \sin{\left(x \right)}$$

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                                                           /       /

$$x^{\sin^{3}{\left(x \right)}} \left(x \left(\left(3 \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{3} \sin^{6}{\left(x \right)} - 3 \left(3 \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \left(3 \log{\left(x \right)} \sin^{2}{\left(x \right)} - 6 \log{\left(x \right)} \cos^{2}{\left(x \right)} - \frac{6 \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) \sin^{3}{\left(x \right)} - 21 \log{\left(x \right)} \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 6 \log{\left(x \right)} \cos^{3}{\left(x \right)} - \frac{9 \sin^{3}{\left(x \right)}}{x} + \frac{18 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{x} - \frac{9 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{2 \sin^{3}{\left(x \right)}}{x^{3}}\right) + 3 \left(\left(3 \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{2} \sin^{3}{\left(x \right)} - 3 \log{\left(x \right)} \sin^{2}{\left(x \right)} + 6 \log{\left(x \right)} \cos^{2}{\left(x \right)} + \frac{6 \sin{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) \sin{\left(x \right)}\right)$$

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Gráfico

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Derivada de x^(sin^3)x

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