Производная (sin(e)^cos(x))

Вы ввели

   cos(x)   
sin      (E)

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

Подробное решение

1. Заменим $u = \cos{\left (x \right )}$.

2. $\frac{d}{d u} \sin^{u}{\left (e \right )} = \log{\left (\sin{\left (e \right )} \right )} \sin^{u}{\left (e \right )}$

3. Затем примените цепочку правил. Умножим на $\frac{d}{d x} \cos{\left (x \right )}$:

   1. Производная косинус есть минус синус:

      $\frac{d}{d x} \cos{\left (x \right )} = - \sin{\left (x \right )}$

   В результате последовательности правил:

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

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Ответ:

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

Первая производная

    cos(x)                      
-sin      (E)*log(sin(E))*sin(x)

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

Вторая производная

   cos(x)    /             2               \            
sin      (E)*\-cos(x) + sin (x)*log(sin(E))/*log(sin(E))

$$\left(\log{\left (\sin{\left (e \right )} \right )} \sin^{2}{\left (x \right )} - \cos{\left (x \right )}\right) \log{\left (\sin{\left (e \right )} \right )} \sin^{\cos{\left (x \right )}}{\left (e \right )}$$

Третья производная

   cos(x)    /       2            2                          \                   
sin      (E)*\1 - log (sin(E))*sin (x) + 3*cos(x)*log(sin(E))/*log(sin(E))*sin(x)

$$\left(- \log^{2}{\left (\sin{\left (e \right )} \right )} \sin^{2}{\left (x \right )} + 3 \log{\left (\sin{\left (e \right )} \right )} \cos{\left (x \right )} + 1\right) \log{\left (\sin{\left (e \right )} \right )} \sin^{\cos{\left (x \right )}}{\left (e \right )} \sin{\left (x \right )}$$