Производная (x/2)^cot(x)

Вы ввели

   cot(x)
/x\      
|-|      
\2/      

$$\left(\frac{x}{2}\right)^{\cot{\left (x \right )}}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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