Производная (tan(x))^(tan(x)+1)

Вы ввели

   tan(x) + 1   
tan          (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

2. Теперь упростим:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

                               /                                                                              2                                                                                                                               2                                                                                                                                               \
                               |                               2                           3     /       2   \                                                                                    /       2   \                  /       2   \                                                            /                 /       2   \                          /       2   \             \|
   1 + tan(x)    /       2   \ |           2      /       2   \  /1 + tan(x)              \    3*\1 + tan (x)/      /       2   \                    2                                          4*\1 + tan (x)/*(1 + tan(x))   2*\1 + tan (x)/ *(1 + tan(x))     /       2   \ /1 + tan(x)              \ |               2*\1 + tan (x)/                          \1 + tan (x)/*(1 + tan(x))||
tan          (x)*\1 + tan (x)/*|12 + 12*tan (x) + \1 + tan (x)/ *|---------- + log(tan(x))|  - ---------------- + 2*\1 + tan (x)/*log(tan(x)) + 4*tan (x)*log(tan(x)) + 4*(1 + tan(x))*tan(x) - ---------------------------- + ----------------------------- + 3*\1 + tan (x)/*|---------- + log(tan(x))|*|2 + 2*tan(x) + --------------- + 2*log(tan(x))*tan(x) - --------------------------||
                               |                                 \  tan(x)                /           2                                                                                                    tan(x)                            3                                 \  tan(x)                / |                    tan(x)                                          2             ||
                               \                                                                   tan (x)                                                                                                                                tan (x)                                                         \                                                                 tan (x)          //

$$\left(\tan^{2}{\left (x \right )} + 1\right) \left(\left(\frac{\tan{\left (x \right )} + 1}{\tan{\left (x \right )}} + \log{\left (\tan{\left (x \right )} \right )}\right)^{3} \left(\tan^{2}{\left (x \right )} + 1\right)^{2} + 3 \left(\frac{\tan{\left (x \right )} + 1}{\tan{\left (x \right )}} + \log{\left (\tan{\left (x \right )} \right )}\right) \left(\tan^{2}{\left (x \right )} + 1\right) \left(- \frac{1}{\tan^{2}{\left (x \right )}} \left(\tan{\left (x \right )} + 1\right) \left(\tan^{2}{\left (x \right )} + 1\right) + \frac{2 \tan^{2}{\left (x \right )} + 2}{\tan{\left (x \right )}} + 2 \log{\left (\tan{\left (x \right )} \right )} \tan{\left (x \right )} + 2 \tan{\left (x \right )} + 2\right) + \frac{2 \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\tan^{3}{\left (x \right )}} \left(\tan{\left (x \right )} + 1\right) - \frac{4}{\tan{\left (x \right )}} \left(\tan{\left (x \right )} + 1\right) \left(\tan^{2}{\left (x \right )} + 1\right) + 4 \left(\tan{\left (x \right )} + 1\right) \tan{\left (x \right )} - \frac{3 \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\tan^{2}{\left (x \right )}} + 2 \left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (\tan{\left (x \right )} \right )} + 4 \log{\left (\tan{\left (x \right )} \right )} \tan^{2}{\left (x \right )} + 12 \tan^{2}{\left (x \right )} + 12\right) \tan^{\tan{\left (x \right )} + 1}{\left (x \right )}$$