Derivada de y=arctg(x)^(x)x

Ha introducido [src]

    x     
atan (x)*x

$$x \operatorname{atan}^{x}{\left(x \right)}$$

atan(x)^x*x

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

    x            x    /       x                       \
atan (x) + x*atan (x)*|---------------- + log(atan(x))|
                      |/     2\                       |
                      \\1 + x /*atan(x)               /

$$x \left(\frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + \log{\left(\operatorname{atan}{\left(x \right)} \right)}\right) \operatorname{atan}^{x}{\left(x \right)} + \operatorname{atan}^{x}{\left(x \right)}$$

Simplificar

Segunda derivada [src]

         /                   /                                              2                    \                   \
         |                   |                                           2*x            x        |                   |
         |                   |                                     -2 + ------ + ----------------|                   |
         |                   |                                 2             2   /     2\        |                   |
    x    |                   |/       x                       \         1 + x    \1 + x /*atan(x)|         2*x       |
atan (x)*|2*log(atan(x)) + x*||---------------- + log(atan(x))|  - ------------------------------| + ----------------|
         |                   ||/     2\                       |           /     2\               |   /     2\        |
         \                   \\\1 + x /*atan(x)               /           \1 + x /*atan(x)       /   \1 + x /*atan(x)/

$$\left(x \left(\left(\frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + \log{\left(\operatorname{atan}{\left(x \right)} \right)}\right)^{2} - \frac{\frac{2 x^{2}}{x^{2} + 1} + \frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} - 2}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}\right) + \frac{2 x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + 2 \log{\left(\operatorname{atan}{\left(x \right)} \right)}\right) \operatorname{atan}^{x}{\left(x \right)}$$

Simplificar

Tercera derivada [src]

         /                                         /                                                          3                                 2                                             /         2                    \\     /         2                    \\
         |                                         |                                               3       8*x            2*x                6*x            /       x                       \ |      2*x            x        ||     |      2*x            x        ||
         |                                         |                                     -8*x - ------- + ------ + ----------------- + ----------------   3*|---------------- + log(atan(x))|*|-2 + ------ + ----------------||   3*|-2 + ------ + ----------------||
         |                                   2     |                                 3          atan(x)        2   /     2\     2      /     2\             |/     2\                       | |          2   /     2\        ||     |          2   /     2\        ||
    x    |  /       x                       \      |/       x                       \                     1 + x    \1 + x /*atan (x)   \1 + x /*atan(x)     \\1 + x /*atan(x)               / \     1 + x    \1 + x /*atan(x)/|     \     1 + x    \1 + x /*atan(x)/|
atan (x)*|3*|---------------- + log(atan(x))|  + x*||---------------- + log(atan(x))|  + -------------------------------------------------------------- - --------------------------------------------------------------------| - ----------------------------------|
         |  |/     2\                       |      ||/     2\                       |                                  2                                                            /     2\                                  |            /     2\                 |
         |  \\1 + x /*atan(x)               /      |\\1 + x /*atan(x)               /                          /     2\                                                             \1 + x /*atan(x)                          |            \1 + x /*atan(x)         |
         \                                         \                                                           \1 + x / *atan(x)                                                                                              /                                     /

$$\left(x \left(\left(\frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + \log{\left(\operatorname{atan}{\left(x \right)} \right)}\right)^{3} - \frac{3 \left(\frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + \log{\left(\operatorname{atan}{\left(x \right)} \right)}\right) \left(\frac{2 x^{2}}{x^{2} + 1} + \frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} - 2\right)}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + \frac{\frac{8 x^{3}}{x^{2} + 1} + \frac{6 x^{2}}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} - 8 x + \frac{2 x}{\left(x^{2} + 1\right) \operatorname{atan}^{2}{\left(x \right)}} - \frac{3}{\operatorname{atan}{\left(x \right)}}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}{\left(x \right)}}\right) + 3 \left(\frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + \log{\left(\operatorname{atan}{\left(x \right)} \right)}\right)^{2} - \frac{3 \left(\frac{2 x^{2}}{x^{2} + 1} + \frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} - 2\right)}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}\right) \operatorname{atan}^{x}{\left(x \right)}$$

Simplificar

Gráfico

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Derivada de y=arctg(x)^(x)x

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