Производная (tan(x))^(x+1)

Вы ввели

   x + 1   
tan     (x)

$$\tan^{x + 1}{\left (x \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(x + 1\right)^{x + 1} \left(\log{\left (x + 1 \right )} + 1\right)$

2. Теперь упростим:

   $\left(x + 1\right)^{x + 1} \left(\log{\left (x + 1 \right )} + 1\right)$

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Ответ:

$\left(x + 1\right)^{x + 1} \left(\log{\left (x + 1 \right )} + 1\right)$

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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