Производная (sinh(x))^(cosh(x))

Вы ввели

    cosh(x)   
sinh       (x)

$$\sinh^{\cosh{\left (x \right )}}{\left (x \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

               /    2                          \
    cosh(x)    |cosh (x)                       |
sinh       (x)*|-------- + log(sinh(x))*sinh(x)|
               \sinh(x)                        /

$$\left(\log{\left (\sinh{\left (x \right )} \right )} \sinh{\left (x \right )} + \frac{\cosh^{2}{\left (x \right )}}{\sinh{\left (x \right )}}\right) \sinh^{\cosh{\left (x \right )}}{\left (x \right )}$$

Вторая производная

               /                                 2                                        \
               |/    2                          \    /        2                  \        |
    cosh(x)    ||cosh (x)                       |    |    cosh (x)               |        |
sinh       (x)*||-------- + log(sinh(x))*sinh(x)|  + |3 - -------- + log(sinh(x))|*cosh(x)|
               |\sinh(x)                        /    |        2                  |        |
               \                                     \    sinh (x)               /        /

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

Третья производная

               /                                 3                                                                                                                                         \
               |/    2                          \                                             2            4        /    2                          \ /        2                  \        |
    cosh(x)    ||cosh (x)                       |                                       2*cosh (x)   2*cosh (x)     |cosh (x)                       | |    cosh (x)               |        |
sinh       (x)*||-------- + log(sinh(x))*sinh(x)|  + 3*sinh(x) + log(sinh(x))*sinh(x) - ---------- + ---------- + 3*|-------- + log(sinh(x))*sinh(x)|*|3 - -------- + log(sinh(x))|*cosh(x)|
               |\sinh(x)                        /                                        sinh(x)          3         \sinh(x)                        / |        2                  |        |
               \                                                                                      sinh (x)                                        \    sinh (x)               /        /

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