Производная atan(e)^(2)^x

Вы ввели

         / x\
         \2 /
(atan(E))    

$$\operatorname{atan}^{2^{x}}{\left (e \right )}$$

Подробное решение

1. Заменим $u = 2^{x}$.

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

3. Затем примените цепочку правил. Умножим на $\frac{d}{d x} 2^{x}$:

   1. $\frac{d}{d x} 2^{x} = 2^{x} \log{\left (2 \right )}$

   В результате последовательности правил:

   $2^{x} \log{\left (2 \right )} \log{\left (\operatorname{atan}{\left (e \right )} \right )} \operatorname{atan}^{2^{x}}{\left (e \right )}$

---

Ответ:

$2^{x} \log{\left (2 \right )} \log{\left (\operatorname{atan}{\left (e \right )} \right )} \operatorname{atan}^{2^{x}}{\left (e \right )}$

Первая производная

            / x\                    
 x          \2 /                    
2 *(atan(E))    *log(2)*log(atan(E))

$$2^{x} \log{\left (2 \right )} \log{\left (\operatorname{atan}{\left (e \right )} \right )} \operatorname{atan}^{2^{x}}{\left (e \right )}$$

Вторая производная

            / x\                                           
 x          \2 /    2    /     x             \             
2 *(atan(E))    *log (2)*\1 + 2 *log(atan(E))/*log(atan(E))

$$2^{x} \left(2^{x} \log{\left (\operatorname{atan}{\left (e \right )} \right )} + 1\right) \log^{2}{\left (2 \right )} \log{\left (\operatorname{atan}{\left (e \right )} \right )} \operatorname{atan}^{2^{x}}{\left (e \right )}$$

Третья производная

            / x\                                                                  
 x          \2 /    3    /     2*x    2               x             \             
2 *(atan(E))    *log (2)*\1 + 2   *log (atan(E)) + 3*2 *log(atan(E))/*log(atan(E))

$$2^{x} \left(2^{2 x} \log^{2}{\left (\operatorname{atan}{\left (e \right )} \right )} + 3 \cdot 2^{x} \log{\left (\operatorname{atan}{\left (e \right )} \right )} + 1\right) \log^{3}{\left (2 \right )} \log{\left (\operatorname{atan}{\left (e \right )} \right )} \operatorname{atan}^{2^{x}}{\left (e \right )}$$