Производная sinh(x)^tanh(x)

Вы ввели

    tanh(x)   
sinh       (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

    tanh(x)    //        2   \                cosh(x)*tanh(x)\
sinh       (x)*|\1 - tanh (x)/*log(sinh(x)) + ---------------|
               \                                  sinh(x)    /

$$\left(\left(- \tanh^{2}{\left (x \right )} + 1\right) \log{\left (\sinh{\left (x \right )} \right )} + \frac{\cosh{\left (x \right )} \tanh{\left (x \right )}}{\sinh{\left (x \right )}}\right) \sinh^{\tanh{\left (x \right )}}{\left (x \right )}$$

Вторая производная

               /                                                2       2                /         2   \                                                           \
    tanh(x)    |//         2   \                cosh(x)*tanh(x)\    cosh (x)*tanh(x)   2*\-1 + tanh (x)/*cosh(x)     /         2   \                               |
sinh       (x)*||\-1 + tanh (x)/*log(sinh(x)) - ---------------|  - ---------------- - ------------------------- + 2*\-1 + tanh (x)/*log(sinh(x))*tanh(x) + tanh(x)|
               |\                                   sinh(x)    /            2                   sinh(x)                                                            |
               \                                                        sinh (x)                                                                                   /

$$\left(\left(\left(\tanh^{2}{\left (x \right )} - 1\right) \log{\left (\sinh{\left (x \right )} \right )} - \frac{\cosh{\left (x \right )} \tanh{\left (x \right )}}{\sinh{\left (x \right )}}\right)^{2} + 2 \left(\tanh^{2}{\left (x \right )} - 1\right) \log{\left (\sinh{\left (x \right )} \right )} \tanh{\left (x \right )} - \frac{2 \cosh{\left (x \right )}}{\sinh{\left (x \right )}} \left(\tanh^{2}{\left (x \right )} - 1\right) + \tanh{\left (x \right )} - \frac{\cosh^{2}{\left (x \right )} \tanh{\left (x \right )}}{\sinh^{2}{\left (x \right )}}\right) \sinh^{\tanh{\left (x \right )}}{\left (x \right )}$$

Третья производная

               /                                                    3                                 2                                                                   /               2                                                         /         2   \        \                                                                       3                    2    /         2   \     /         2   \                \
    tanh(x)    |    //         2   \                cosh(x)*tanh(x)\          2        /         2   \                   //         2   \                cosh(x)*tanh(x)\ |           cosh (x)*tanh(x)     /         2   \                        2*\-1 + tanh (x)/*cosh(x)|         2    /         2   \                2*cosh(x)*tanh(x)   2*cosh (x)*tanh(x)   3*cosh (x)*\-1 + tanh (x)/   6*\-1 + tanh (x)/*cosh(x)*tanh(x)|
sinh       (x)*|3 - |\-1 + tanh (x)/*log(sinh(x)) - ---------------|  - 3*tanh (x) - 2*\-1 + tanh (x)/ *log(sinh(x)) + 3*|\-1 + tanh (x)/*log(sinh(x)) - ---------------|*|-tanh(x) + ---------------- - 2*\-1 + tanh (x)/*log(sinh(x))*tanh(x) + -------------------------| - 4*tanh (x)*\-1 + tanh (x)/*log(sinh(x)) - ----------------- + ------------------ + -------------------------- + ---------------------------------|
               |    \                                   sinh(x)    /                                                     \                                   sinh(x)    / |                   2                                                            sinh(x)         |                                                  sinh(x)                 3                        2                            sinh(x)             |
               \                                                                                                                                                          \               sinh (x)                                                                         /                                                                      sinh (x)                 sinh (x)                                             /

$$\left(- \left(\left(\tanh^{2}{\left (x \right )} - 1\right) \log{\left (\sinh{\left (x \right )} \right )} - \frac{\cosh{\left (x \right )} \tanh{\left (x \right )}}{\sinh{\left (x \right )}}\right)^{3} + 3 \left(\left(\tanh^{2}{\left (x \right )} - 1\right) \log{\left (\sinh{\left (x \right )} \right )} - \frac{\cosh{\left (x \right )} \tanh{\left (x \right )}}{\sinh{\left (x \right )}}\right) \left(- 2 \left(\tanh^{2}{\left (x \right )} - 1\right) \log{\left (\sinh{\left (x \right )} \right )} \tanh{\left (x \right )} + \frac{2 \cosh{\left (x \right )}}{\sinh{\left (x \right )}} \left(\tanh^{2}{\left (x \right )} - 1\right) - \tanh{\left (x \right )} + \frac{\cosh^{2}{\left (x \right )} \tanh{\left (x \right )}}{\sinh^{2}{\left (x \right )}}\right) - 2 \left(\tanh^{2}{\left (x \right )} - 1\right)^{2} \log{\left (\sinh{\left (x \right )} \right )} - 4 \left(\tanh^{2}{\left (x \right )} - 1\right) \log{\left (\sinh{\left (x \right )} \right )} \tanh^{2}{\left (x \right )} + \frac{6 \cosh{\left (x \right )}}{\sinh{\left (x \right )}} \left(\tanh^{2}{\left (x \right )} - 1\right) \tanh{\left (x \right )} + \frac{3 \cosh^{2}{\left (x \right )}}{\sinh^{2}{\left (x \right )}} \left(\tanh^{2}{\left (x \right )} - 1\right) - 3 \tanh^{2}{\left (x \right )} + 3 - \frac{2 \cosh{\left (x \right )}}{\sinh{\left (x \right )}} \tanh{\left (x \right )} + \frac{2 \cosh^{3}{\left (x \right )}}{\sinh^{3}{\left (x \right )}} \tanh{\left (x \right )}\right) \sinh^{\tanh{\left (x \right )}}{\left (x \right )}$$