Sr Examen

Derivada de y=(arctgx)^sinx

Función f() - derivada -er orden en el punto
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Gráfico:

interior superior

Definida a trozos:

Solución

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

    Perola derivada


Respuesta:

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