Derivada de x*x^sen(x)

Solución

Ha introducido [src]

   sin(x)
x*x

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

x*x^sin(x)

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

        /                          2     /                          3                                                                                                                  \                                        \
 sin(x) |  /sin(x)                \      |  /sin(x)                \                    2*sin(x)   3*sin(x)   3*cos(x)     /sin(x)                \ /sin(x)                   2*cos(x)\|   3*sin(x)                     6*cos(x)|
x      *|3*|------ + cos(x)*log(x)|  - x*|- |------ + cos(x)*log(x)|  + cos(x)*log(x) - -------- + -------- + -------- + 3*|------ + cos(x)*log(x)|*|------ + log(x)*sin(x) - --------|| - -------- - 3*log(x)*sin(x) + --------|
        |  \  x                   /      |  \  x                   /                        3         x           2        \  x                   / |   2                        x    ||       2                           x    |
        \                                \                                                 x                     x                                  \  x                              //      x                                 /

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

Simplificar

Gráfico

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Derivada de x*x^sen(x)

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