Производная (cot(x)^7)^atan(sqrt(x))

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↑ Функция f () ? - производная -го порядка

Решение

Вы ввели
[LaTeX]
           /  ___\
       atan\\/ x /
   7              
cot (x)           
$$\left(\cot^{7}{\left (x \right )}\right)^{\operatorname{atan}{\left (\sqrt{x} \right )}}$$
Подробное решение
[LaTeX]
  1. Не могу найти шаги в поиске этой производной.

    Но производная


Ответ:

График
[LaTeX]
Первая производная
[LaTeX]
           /  ___\                                                 
       atan\\/ x / //          2   \     /  ___\        /   7   \ \
   7               |\-7 - 7*cot (x)/*atan\\/ x /     log\cot (x)/ |
cot (x)           *|---------------------------- + ---------------|
                   |           cot(x)                  ___        |
                   \                               2*\/ x *(1 + x)/
$$\left(\frac{\operatorname{atan}{\left (\sqrt{x} \right )}}{\cot{\left (x \right )}} \left(- 7 \cot^{2}{\left (x \right )} - 7\right) + \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{2 \sqrt{x} \left(x + 1\right)}\right) \left(\cot^{7}{\left (x \right )}\right)^{\operatorname{atan}{\left (\sqrt{x} \right )}}$$
Вторая производная
[LaTeX]
                   /                                                2                                                                                                                         \
                   |/      /   7   \      /       2   \     /  ___\\                                                                                                                          |
                   ||   log\cot (x)/   14*\1 + cot (x)/*atan\\/ x /|                                                                                                                          |
           /  ___\ ||- ------------- + ----------------------------|                                                  2                                                                       |
       atan\\/ x / ||    ___                      cot(x)           |                                     /       2   \      /  ___\        /   7   \         /   7   \        /       2   \   |
   7               |\  \/ x *(1 + x)                               /       /       2   \     /  ___\   7*\1 + cot (x)/ *atan\\/ x /     log\cot (x)/      log\cot (x)/      7*\1 + cot (x)/   |
cot (x)           *|------------------------------------------------- + 14*\1 + cot (x)/*atan\\/ x / - ---------------------------- - ---------------- - -------------- - --------------------|
                   |                        4                                                                       2                     ___        2      3/2             ___               |
                   \                                                                                             cot (x)              2*\/ x *(1 + x)    4*x   *(1 + x)   \/ x *(1 + x)*cot(x)/
$$\left(\frac{1}{4} \left(\frac{14 \operatorname{atan}{\left (\sqrt{x} \right )}}{\cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) - \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{\sqrt{x} \left(x + 1\right)}\right)^{2} - \frac{7 \left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{\cot^{2}{\left (x \right )}} \operatorname{atan}{\left (\sqrt{x} \right )} + 14 \left(\cot^{2}{\left (x \right )} + 1\right) \operatorname{atan}{\left (\sqrt{x} \right )} - \frac{7 \cot^{2}{\left (x \right )} + 7}{\sqrt{x} \left(x + 1\right) \cot{\left (x \right )}} - \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{2 \sqrt{x} \left(x + 1\right)^{2}} - \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{4 x^{\frac{3}{2}} \left(x + 1\right)}\right) \left(\cot^{7}{\left (x \right )}\right)^{\operatorname{atan}{\left (\sqrt{x} \right )}}$$
Третья производная
[LaTeX]
                   /                                                  3                                                      /                                                                                 2                                   \                                                                                                                                                                                                                                                         \
                   |  /      /   7   \      /       2   \     /  ___\\      /      /   7   \      /       2   \     /  ___\\ |                                    /   7   \        /   7   \      /       2   \      /  ___\        /       2   \  |                                                                                                                                                                                                                                                         |
                   |  |   log\cot (x)/   14*\1 + cot (x)/*atan\\/ x /|      |   log\cot (x)/   14*\1 + cot (x)/*atan\\/ x /| |     /       2   \     /  ___\   log\cot (x)/   2*log\cot (x)/   28*\1 + cot (x)/ *atan\\/ x /     28*\1 + cot (x)/  |                                                                                                                                                                                                                                                         |
           /  ___\ |  |- ------------- + ----------------------------|    3*|- ------------- + ----------------------------|*|- 56*\1 + cot (x)/*atan\\/ x / + ------------ + -------------- + ----------------------------- + --------------------|                                                                                            3                                                  2                                                   2                                                     |
       atan\\/ x / |  |    ___                      cot(x)           |      |    ___                      cot(x)           | |                                  3/2             ___        2                 2                   ___               |       /   7   \         /   7   \                                             /       2   \      /  ___\      /       2   \      /       2   \      /  ___\        /   7   \         /       2   \              /       2   \            /       2   \  |
   7               |  \  \/ x *(1 + x)                               /      \  \/ x *(1 + x)                               / \                                 x   *(1 + x)   \/ x *(1 + x)               cot (x)              \/ x *(1 + x)*cot(x)/    log\cot (x)/      log\cot (x)/       /       2   \     /  ___\          14*\1 + cot (x)/ *atan\\/ x /   21*\1 + cot (x)/   28*\1 + cot (x)/ *atan\\/ x /   3*log\cot (x)/      21*\1 + cot (x)/           21*\1 + cot (x)/         21*\1 + cot (x)/  |
cot (x)           *|- ------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------- + --------------- - 28*\1 + cot (x)/*atan\\/ x /*cot(x) - ----------------------------- + ---------------- + ----------------------------- + -------------- - ----------------------- + ----------------------- + ---------------------|
                   |                          8                                                                                                               8                                                                                          ___        3      3/2        2                                                       3                    ___                         cot(x)                 5/2               ___            2          ___        2             3/2               |
                   \                                                                                                                                                                                                                                   \/ x *(1 + x)    2*x   *(1 + x)                                                     cot (x)               \/ x *(1 + x)                                     8*x   *(1 + x)   2*\/ x *(1 + x)*cot (x)   2*\/ x *(1 + x) *cot(x)   4*x   *(1 + x)*cot(x)/
$$\left(- \frac{1}{8} \left(\frac{14 \operatorname{atan}{\left (\sqrt{x} \right )}}{\cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) - \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{\sqrt{x} \left(x + 1\right)}\right)^{3} + \frac{3}{8} \left(\frac{14 \operatorname{atan}{\left (\sqrt{x} \right )}}{\cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) - \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{\sqrt{x} \left(x + 1\right)}\right) \left(\frac{28 \left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{\cot^{2}{\left (x \right )}} \operatorname{atan}{\left (\sqrt{x} \right )} - 56 \left(\cot^{2}{\left (x \right )} + 1\right) \operatorname{atan}{\left (\sqrt{x} \right )} + \frac{28 \cot^{2}{\left (x \right )} + 28}{\sqrt{x} \left(x + 1\right) \cot{\left (x \right )}} + \frac{2 \log{\left (\cot^{7}{\left (x \right )} \right )}}{\sqrt{x} \left(x + 1\right)^{2}} + \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{x^{\frac{3}{2}} \left(x + 1\right)}\right) - \frac{14 \left(\cot^{2}{\left (x \right )} + 1\right)^{3}}{\cot^{3}{\left (x \right )}} \operatorname{atan}{\left (\sqrt{x} \right )} + \frac{28 \left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{\cot{\left (x \right )}} \operatorname{atan}{\left (\sqrt{x} \right )} - 28 \left(\cot^{2}{\left (x \right )} + 1\right) \cot{\left (x \right )} \operatorname{atan}{\left (\sqrt{x} \right )} - \frac{21 \left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{2 \sqrt{x} \left(x + 1\right) \cot^{2}{\left (x \right )}} + \frac{21 \cot^{2}{\left (x \right )} + 21}{\sqrt{x} \left(x + 1\right)} + \frac{21 \cot^{2}{\left (x \right )} + 21}{2 \sqrt{x} \left(x + 1\right)^{2} \cot{\left (x \right )}} + \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{\sqrt{x} \left(x + 1\right)^{3}} + \frac{21 \cot^{2}{\left (x \right )} + 21}{4 x^{\frac{3}{2}} \left(x + 1\right) \cot{\left (x \right )}} + \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{2 x^{\frac{3}{2}} \left(x + 1\right)^{2}} + \frac{3 \log{\left (\cot^{7}{\left (x \right )} \right )}}{8 x^{\frac{5}{2}} \left(x + 1\right)}\right) \left(\cot^{7}{\left (x \right )}\right)^{\operatorname{atan}{\left (\sqrt{x} \right )}}$$