Производная 2^cot(x)^(23)

Вы ввели

    23   
 cot  (x)
2        

$$2^{\cot^{23}{\left (x \right )}}$$

Подробное решение

1. Заменим $u = \cot^{23}{\left (x \right )}$.

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

3. Затем примените цепочку правил. Умножим на $\frac{d}{d x} \cot^{23}{\left (x \right )}$:

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

   2. В силу правила, применим: $u^{23}$ получим $23 u^{22}$

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

      1. Есть несколько способов вычислить эту производную.

         Один из способов:

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

      В результате последовательности правил:

      $- \frac{23 \left(\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}\right) \cot^{22}{\left (x \right )}}{\cos^{2}{\left (x \right )} \tan^{2}{\left (x \right )}}$

   В результате последовательности правил:

   $- \frac{23 \cdot 2^{\cot^{23}{\left (x \right )}} \log{\left (2 \right )} \cot^{22}{\left (x \right )}}{\cos^{2}{\left (x \right )} \tan^{2}{\left (x \right )}} \left(\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}\right)$

4. Теперь упростим:

   $- \frac{2^{\frac{1}{\tan^{23}{\left (x \right )}}} \cos^{22}{\left (x \right )}}{\sin^{24}{\left (x \right )}} \log{\left (8388608 \right )}$

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Ответ:

$- \frac{2^{\frac{1}{\tan^{23}{\left (x \right )}}} \cos^{22}{\left (x \right )}}{\sin^{24}{\left (x \right )}} \log{\left (8388608 \right )}$

Первая производная

    23                                      
 cot  (x)    22    /            2   \       
2        *cot  (x)*\-23 - 23*cot (x)/*log(2)

$$2^{\cot^{23}{\left (x \right )}} \left(- 23 \cot^{2}{\left (x \right )} - 23\right) \log{\left (2 \right )} \cot^{22}{\left (x \right )}$$

Вторая производная

       23                                                                                      
    cot  (x)    21    /       2   \ /           2            23    /       2   \       \       
23*2        *cot  (x)*\1 + cot (x)/*\22 + 24*cot (x) + 23*cot  (x)*\1 + cot (x)/*log(2)/*log(2)

$$23 \cdot 2^{\cot^{23}{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) \left(23 \left(\cot^{2}{\left (x \right )} + 1\right) \log{\left (2 \right )} \cot^{23}{\left (x \right )} + 24 \cot^{2}{\left (x \right )} + 22\right) \log{\left (2 \right )} \cot^{21}{\left (x \right )}$$

Третья производная

        23                           /                             2                                                                                    2                                      2                \       
     cot  (x)    20    /       2   \ |     4          /       2   \           2    /       2   \          25    /       2   \              /       2   \     46       2           /       2   \     23          |       
-23*2        *cot  (x)*\1 + cot (x)/*\4*cot (x) + 462*\1 + cot (x)/  + 134*cot (x)*\1 + cot (x)/ + 138*cot  (x)*\1 + cot (x)/*log(2) + 529*\1 + cot (x)/ *cot  (x)*log (2) + 1518*\1 + cot (x)/ *cot  (x)*log(2)/*log(2)

$$- 23 \cdot 2^{\cot^{23}{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) \left(529 \left(\cot^{2}{\left (x \right )} + 1\right)^{2} \log^{2}{\left (2 \right )} \cot^{46}{\left (x \right )} + 1518 \left(\cot^{2}{\left (x \right )} + 1\right)^{2} \log{\left (2 \right )} \cot^{23}{\left (x \right )} + 462 \left(\cot^{2}{\left (x \right )} + 1\right)^{2} + 138 \left(\cot^{2}{\left (x \right )} + 1\right) \log{\left (2 \right )} \cot^{25}{\left (x \right )} + 134 \left(\cot^{2}{\left (x \right )} + 1\right) \cot^{2}{\left (x \right )} + 4 \cot^{4}{\left (x \right )}\right) \log{\left (2 \right )} \cot^{20}{\left (x \right )}$$