Производная 5^(log(tan(x)))

Вы ввели

 log(tan(x))
5           

$$5^{\log{\left (\tan{\left (x \right )} \right )}}$$

Подробное решение

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

2. $\frac{d}{d u} 5^{u} = 5^{u} \log{\left (5 \right )}$

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

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

   2. Производная $\log{\left (u \right )}$ является $\frac{1}{u}$.

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

      1. Есть несколько способов вычислить эту производную.

         Один из способов:

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

      В результате последовательности правил:

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

   В результате последовательности правил:

   $\frac{5^{\log{\left (\tan{\left (x \right )} \right )}} \log{\left (5 \right )}}{\cos^{2}{\left (x \right )} \tan{\left (x \right )}} \left(\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}\right)$

4. Теперь упростим:

   $\frac{5^{\log{\left (\tan{\left (x \right )} \right )}}}{\sin{\left (2 x \right )}} \log{\left (25 \right )}$

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Ответ:

$\frac{5^{\log{\left (\tan{\left (x \right )} \right )}}}{\sin{\left (2 x \right )}} \log{\left (25 \right )}$

Первая производная

 log(tan(x)) /       2   \       
5           *\1 + tan (x)/*log(5)
---------------------------------
              tan(x)             

$$\frac{5^{\log{\left (\tan{\left (x \right )} \right )}}}{\tan{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (5 \right )}$$

Вторая производная

                           /           2      /       2   \       \       
 log(tan(x)) /       2   \ |    1 + tan (x)   \1 + tan (x)/*log(5)|       
5           *\1 + tan (x)/*|2 - ----------- + --------------------|*log(5)
                           |         2                 2          |       
                           \      tan (x)           tan (x)       /       

$$5^{\log{\left (\tan{\left (x \right )} \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) \left(- \frac{\tan^{2}{\left (x \right )} + 1}{\tan^{2}{\left (x \right )}} + \frac{\log{\left (5 \right )}}{\tan^{2}{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) + 2\right) \log{\left (5 \right )}$$

Третья производная

                           /                                            2                2                          2                                \       
                           |             /       2   \     /       2   \    /       2   \     2        /       2   \             /       2   \       |       
 log(tan(x)) /       2   \ |           4*\1 + tan (x)/   2*\1 + tan (x)/    \1 + tan (x)/ *log (5)   3*\1 + tan (x)/ *log(5)   6*\1 + tan (x)/*log(5)|       
5           *\1 + tan (x)/*|4*tan(x) - --------------- + ---------------- + ---------------------- - ----------------------- + ----------------------|*log(5)
                           |                tan(x)              3                     3                         3                      tan(x)        |       
                           \                                 tan (x)               tan (x)                   tan (x)                                 /       

$$5^{\log{\left (\tan{\left (x \right )} \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) \left(- \frac{3 \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\tan^{3}{\left (x \right )}} \log{\left (5 \right )} + \frac{2 \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\tan^{3}{\left (x \right )}} + \frac{\left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\tan^{3}{\left (x \right )}} \log^{2}{\left (5 \right )} - \frac{4 \tan^{2}{\left (x \right )} + 4}{\tan{\left (x \right )}} + \frac{6 \log{\left (5 \right )}}{\tan{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) + 4 \tan{\left (x \right )}\right) \log{\left (5 \right )}$$