Производная (cos(x)^sin(x))

Вы ввели

   sin(x)   
cos      (x)

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

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

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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

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