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y=arctg(x)^(x)x
Derivada de la función/ tg(x)/ arctg/ (x)^(x)/ y=arctg(x)^(x)x

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

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

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src] 
    x     
atan (x)*x
xatan⁡x(x)x \operatorname{atan}^{x}{\left(x \right)} 
atan(x)^x*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)=atan⁡x(x)f{\left(x \right)} = \operatorname{atan}^{x}{\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

      xx(log⁡(x)+1)x^{x} \left(\log{\left(x \right)} + 1\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: xxx(log⁡(x)+1)+atan⁡x(x)x x^{x} \left(\log{\left(x \right)} + 1\right) + \operatorname{atan}^{x}{\left(x \right)}

  2. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-500500
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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(x(x2+1)atan⁡(x)+log⁡(atan⁡(x)))atan⁡x(x)+atan⁡x(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)/
(x((x(x2+1)atan⁡(x)+log⁡(atan⁡(x)))2−2x2x2+1+x(x2+1)atan⁡(x)−2(x2+1)atan⁡(x))+2x(x2+1)atan⁡(x)+2log⁡(atan⁡(x)))atan⁡x(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)                                                                                              /                                     /
(x((x(x2+1)atan⁡(x)+log⁡(atan⁡(x)))3−3(x(x2+1)atan⁡(x)+log⁡(atan⁡(x)))(2x2x2+1+x(x2+1)atan⁡(x)−2)(x2+1)atan⁡(x)+8x3x2+1+6x2(x2+1)atan⁡(x)−8x+2x(x2+1)atan⁡2(x)−3atan⁡(x)(x2+1)2atan⁡(x))+3(x(x2+1)atan⁡(x)+log⁡(atan⁡(x)))2−3(2x2x2+1+x(x2+1)atan⁡(x)−2)(x2+1)atan⁡(x))atan⁡x(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)}
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Gráfico
Derivada de y=arctg(x)^(x)x
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