Производная (cos(5*x))^(e^x)

Вы ввели

          / x\
          \E /
(cos(5*x))    

$$\cos^{e^{x}}{\left (5 x \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

2. Теперь упростим:

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

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Ответ:

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

Первая производная

          / x\ /                      x         \
          \E / | x                 5*e *sin(5*x)|
(cos(5*x))    *|e *log(cos(5*x)) - -------------|
               \                      cos(5*x)  /

$$\left(e^{x} \log{\left (\cos{\left (5 x \right )} \right )} - \frac{5 e^{x} \sin{\left (5 x \right )}}{\cos{\left (5 x \right )}}\right) \cos^{e^{x}}{\left (5 x \right )}$$

Вторая производная

          / x\ /                                    2            2                                   \   
          \e / |      /  5*sin(5*x)                \   x   25*sin (5*x)   10*sin(5*x)                |  x
(cos(5*x))    *|-25 + |- ---------- + log(cos(5*x))| *e  - ------------ - ----------- + log(cos(5*x))|*e 
               |      \   cos(5*x)                 /           2            cos(5*x)                 |   
               \                                            cos (5*x)                                /   

$$\left(\left(\log{\left (\cos{\left (5 x \right )} \right )} - \frac{5 \sin{\left (5 x \right )}}{\cos{\left (5 x \right )}}\right)^{2} e^{x} + \log{\left (\cos{\left (5 x \right )} \right )} - \frac{25 \sin^{2}{\left (5 x \right )}}{\cos^{2}{\left (5 x \right )}} - \frac{10 \sin{\left (5 x \right )}}{\cos{\left (5 x \right )}} - 25\right) e^{x} \cos^{e^{x}}{\left (5 x \right )}$$

Третья производная

          / x\ /                                    3                              3              2                                         /                                         2     \                   \   
          \e / |      /  5*sin(5*x)                \   2*x   265*sin(5*x)   250*sin (5*x)   75*sin (5*x)     /  5*sin(5*x)                \ |                     10*sin(5*x)   25*sin (5*x)|  x                |  x
(cos(5*x))    *|-75 + |- ---------- + log(cos(5*x))| *e    - ------------ - ------------- - ------------ - 3*|- ---------- + log(cos(5*x))|*|25 - log(cos(5*x)) + ----------- + ------------|*e  + log(cos(5*x))|*e 
               |      \   cos(5*x)                 /           cos(5*x)          3              2            \   cos(5*x)                 / |                       cos(5*x)        2       |                   |   
               \                                                              cos (5*x)      cos (5*x)                                      \                                    cos (5*x)  /                   /   

$$\left(\left(\log{\left (\cos{\left (5 x \right )} \right )} - \frac{5 \sin{\left (5 x \right )}}{\cos{\left (5 x \right )}}\right)^{3} e^{2 x} - 3 \left(\log{\left (\cos{\left (5 x \right )} \right )} - \frac{5 \sin{\left (5 x \right )}}{\cos{\left (5 x \right )}}\right) \left(- \log{\left (\cos{\left (5 x \right )} \right )} + \frac{25 \sin^{2}{\left (5 x \right )}}{\cos^{2}{\left (5 x \right )}} + \frac{10 \sin{\left (5 x \right )}}{\cos{\left (5 x \right )}} + 25\right) e^{x} + \log{\left (\cos{\left (5 x \right )} \right )} - \frac{250 \sin^{3}{\left (5 x \right )}}{\cos^{3}{\left (5 x \right )}} - \frac{75 \sin^{2}{\left (5 x \right )}}{\cos^{2}{\left (5 x \right )}} - \frac{265 \sin{\left (5 x \right )}}{\cos{\left (5 x \right )}} - 75\right) e^{x} \cos^{e^{x}}{\left (5 x \right )}$$