Sr Examen

Derivada de y=x^arctg(x)

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

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src]
 atan(x)
x       
$$x^{\operatorname{atan}{\left(x \right)}}$$
x^atan(x)
Solución detallada
  1. No logro encontrar los pasos en la búsqueda de esta derivada.

    Perola derivada


Respuesta:

Gráfica
Primera derivada [src]
 atan(x) /atan(x)   log(x)\
x       *|------- + ------|
         |   x           2|
         \          1 + x /
$$x^{\operatorname{atan}{\left(x \right)}} \left(\frac{\log{\left(x \right)}}{x^{2} + 1} + \frac{\operatorname{atan}{\left(x \right)}}{x}\right)$$
Segunda derivada [src]
         /                  2                                    \
 atan(x) |/atan(x)   log(x)\    atan(x)       2        2*x*log(x)|
x       *||------- + ------|  - ------- + ---------- - ----------|
         ||   x           2|        2       /     2\           2 |
         |\          1 + x /       x      x*\1 + x /   /     2\  |
         \                                             \1 + x /  /
$$x^{\operatorname{atan}{\left(x \right)}} \left(- \frac{2 x \log{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \left(\frac{\log{\left(x \right)}}{x^{2} + 1} + \frac{\operatorname{atan}{\left(x \right)}}{x}\right)^{2} + \frac{2}{x \left(x^{2} + 1\right)} - \frac{\operatorname{atan}{\left(x \right)}}{x^{2}}\right)$$
Tercera derivada [src]
         /                  3                                                                                                                   2       \
 atan(x) |/atan(x)   log(x)\        6            3          /atan(x)   log(x)\ /atan(x)       2        2*x*log(x)\    2*log(x)   2*atan(x)   8*x *log(x)|
x       *||------- + ------|  - --------- - ----------- - 3*|------- + ------|*|------- - ---------- + ----------| - --------- + --------- + -----------|
         ||   x           2|            2    2 /     2\     |   x           2| |    2       /     2\           2 |           2        3               3 |
         |\          1 + x /    /     2\    x *\1 + x /     \          1 + x / |   x      x*\1 + x /   /     2\  |   /     2\        x        /     2\  |
         \                      \1 + x /                                       \                       \1 + x /  /   \1 + x /                 \1 + x /  /
$$x^{\operatorname{atan}{\left(x \right)}} \left(\frac{8 x^{2} \log{\left(x \right)}}{\left(x^{2} + 1\right)^{3}} + \left(\frac{\log{\left(x \right)}}{x^{2} + 1} + \frac{\operatorname{atan}{\left(x \right)}}{x}\right)^{3} - 3 \left(\frac{\log{\left(x \right)}}{x^{2} + 1} + \frac{\operatorname{atan}{\left(x \right)}}{x}\right) \left(\frac{2 x \log{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} - \frac{2}{x \left(x^{2} + 1\right)} + \frac{\operatorname{atan}{\left(x \right)}}{x^{2}}\right) - \frac{2 \log{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} - \frac{6}{\left(x^{2} + 1\right)^{2}} - \frac{3}{x^{2} \left(x^{2} + 1\right)} + \frac{2 \operatorname{atan}{\left(x \right)}}{x^{3}}\right)$$
Gráfico
Derivada de y=x^arctg(x)