Производная (log(t))^sin(t)

Вы ввели

   sin(t)   
log      (t)

$$\log^{\sin{\left (t \right )}}{\left (t \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

   sin(t)    /                      sin(t) \
log      (t)*|cos(t)*log(log(t)) + --------|
             \                     t*log(t)/

$$\left(\log{\left (\log{\left (t \right )} \right )} \cos{\left (t \right )} + \frac{\sin{\left (t \right )}}{t \log{\left (t \right )}}\right) \log^{\sin{\left (t \right )}}{\left (t \right )}$$

Вторая производная

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

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

Третья производная

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

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