Производная cos(2)^(3)^x

$$cos^{3^{x}}{left (2 right )}$$

1. Заменим
   u = 3^{x}
   .

2. frac{d}{d u} cos^{u}{left (2 right )} = left(log{left (- cos{left (2 right )} right )} + i piright) cos^{u}{left (2 right )}

3. Затем примените цепочку правил. Умножим на
   frac{d}{d x} 3^{x}
   :

   1. frac{d}{d x} 3^{x} = 3^{x} log{left (3 right )}

   В результате последовательности правил:

   3^{x} left(log{left (- cos{left (2 right )} right )} + i piright) log{left (3 right )} cos^{3^{x}}{left (2 right )}

---

Ответ:

3^{x} left(log{left (- cos{left (2 right )} right )} + i piright) log{left (3 right )} cos^{3^{x}}{left (2 right )}

/ x
x 3 /
3 *(cos(2)) *(pi*I + log(-cos(2)))*log(3)

$$3^{x} left(log{left (- cos{left (2 right )} right )} + i piright) log{left (3 right )} cos^{3^{x}}{left (2 right )}$$

Вторая производная

/ x
x 3 / 2 / x
3 *(cos(2)) *log (3)*1 + 3 *(pi*I + log(-cos(2)))/*(pi*I + log(-cos(2)))

$$3^{x} left(3^{x} left(log{left (- cos{left (2 right )} right )} + i piright) + 1right) left(log{left (- cos{left (2 right )} right )} + i piright) log^{2}{left (3 right )} cos^{3^{x}}{left (2 right )}$$

Третья производная

/ x
x 3 / 3 / 2*x 2 x
3 *(cos(2)) *log (3)*(pi*I + log(-cos(2)))*1 + 3 *(pi*I + log(-cos(2))) + 3*3 *(pi*I + log(-cos(2)))/

$$3^{x} left(log{left (- cos{left (2 right )} right )} + i piright) left(3^{2 x} left(log{left (- cos{left (2 right )} right )} + i piright)^{2} + 3 cdot 3^{x} left(log{left (- cos{left (2 right )} right )} + i piright) + 1right) log^{3}{left (3 right )} cos^{3^{x}}{left (2 right )}$$