Производная cot(x)^cos(x)

Вы ввели

   cos(x)   
cot      (x)

$$\cot^{\cos{\left(x \right)}}{\left(x \right)}$$

d /   cos(x)   \
--\cot      (x)/
dx              

$$\frac{d}{d x} \cot^{\cos{\left(x \right)}}{\left(x \right)}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

             /                      /        2   \       \
   cos(x)    |                      \-1 - cot (x)/*cos(x)|
cot      (x)*|-log(cot(x))*sin(x) + ---------------------|
             \                              cot(x)       /

$$\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \cos{\left(x \right)}}{\cot{\left(x \right)}} - \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)}\right) \cot^{\cos{\left(x \right)}}{\left(x \right)}$$

Вторая производная

             /                                           2                                                              2                                \
             |/                     /       2   \       \                                                  /       2   \             /       2   \       |
   cos(x)    ||                     \1 + cot (x)/*cos(x)|                           /       2   \          \1 + cot (x)/ *cos(x)   2*\1 + cot (x)/*sin(x)|
cot      (x)*||log(cot(x))*sin(x) + --------------------|  - cos(x)*log(cot(x)) + 2*\1 + cot (x)/*cos(x) - --------------------- + ----------------------|
             |\                            cot(x)       /                                                            2                     cot(x)        |
             \                                                                                                    cot (x)                                /

$$\left(\left(\frac{\left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)}\right)^{2} - \frac{\left(\cot^{2}{\left(x \right)} + 1\right)^{2} \cos{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} - \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cot^{\cos{\left(x \right)}}{\left(x \right)}$$

Третья производная

             /                                             3                                                                                               /                                                           2                                \                                                  3                         2                                                  2       \
             |  /                     /       2   \       \                                                    /                     /       2   \       \ |                                              /       2   \             /       2   \       |                                     /       2   \             /       2   \             /       2   \            /       2   \        |
   cos(x)    |  |                     \1 + cot (x)/*cos(x)|                           /       2   \            |                     \1 + cot (x)/*cos(x)| |                       /       2   \          \1 + cot (x)/ *cos(x)   2*\1 + cot (x)/*sin(x)|     /       2   \                 2*\1 + cot (x)/ *cos(x)   3*\1 + cot (x)/ *sin(x)   3*\1 + cot (x)/*cos(x)   4*\1 + cot (x)/ *cos(x)|
cot      (x)*|- |log(cot(x))*sin(x) + --------------------|  + log(cot(x))*sin(x) - 6*\1 + cot (x)/*sin(x) + 3*|log(cot(x))*sin(x) + --------------------|*|cos(x)*log(cot(x)) - 2*\1 + cot (x)/*cos(x) + --------------------- - ----------------------| - 4*\1 + cot (x)/*cos(x)*cot(x) - ----------------------- + ----------------------- + ---------------------- + -----------------------|
             |  \                            cot(x)       /                                                    \                            cot(x)       / |                                                        2                     cot(x)        |                                              3                         2                      cot(x)                    cot(x)        |
             \                                                                                                                                             \                                                     cot (x)                                /                                           cot (x)                   cot (x)                                                           /

$$\left(- \left(\frac{\left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)}\right)^{3} + 3 \left(\frac{\left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)}\right) \left(\frac{\left(\cot^{2}{\left(x \right)} + 1\right)^{2} \cos{\left(x \right)}}{\cot^{2}{\left(x \right)}} - \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cot{\left(x \right)}} - 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}\right) - \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)^{3} \cos{\left(x \right)}}{\cot^{3}{\left(x \right)}} + \frac{3 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\cot^{2}{\left(x \right)}} + \frac{4 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \cos{\left(x \right)}}{\cot{\left(x \right)}} - 6 \left(\cot^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} - 4 \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \cot{\left(x \right)} + \frac{3 \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \sin{\left(x \right)}\right) \cot^{\cos{\left(x \right)}}{\left(x \right)}$$

График