Производная sin(x)^cos(x)

Вы ввели

   cos(x)   
sin      (x)

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

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

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

             /                              2                                     \
             |/                        2   \    /       2                 \       |
   cos(x)    ||                     cos (x)|    |    cos (x)              |       |
sin      (x)*||log(sin(x))*sin(x) - -------|  - |3 + ------- + log(sin(x))|*cos(x)|
             |\                      sin(x)/    |       2                 |       |
             \                                  \    sin (x)              /       /

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

Третья производная

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

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

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