Производная (2+log(x))^sin(x)

Вы ввели

            sin(x)
(2 + log(x))      

$$\left(\log{\left(x \right)} + 2\right)^{\sin{\left(x \right)}}$$

d /            sin(x)\
--\(2 + log(x))      /
dx                    

$$\frac{d}{d x} \left(\log{\left(x \right)} + 2\right)^{\sin{\left(x \right)}}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

            sin(x) /                             sin(x)    \
(2 + log(x))      *|cos(x)*log(2 + log(x)) + --------------|
                   \                         x*(2 + log(x))/

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

Вторая производная

                   /                                         2                                                                               \
            sin(x) |/                             sin(x)    \                                  sin(x)            sin(x)           2*cos(x)   |
(2 + log(x))      *||cos(x)*log(2 + log(x)) + --------------|  - log(2 + log(x))*sin(x) - --------------- - ---------------- + --------------|
                   |\                         x*(2 + log(x))/                              2                 2             2   x*(2 + log(x))|
                   \                                                                      x *(2 + log(x))   x *(2 + log(x))                  /

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

Третья производная

                   /                                         3                                                                                                                                                                                                                                                                    \
            sin(x) |/                             sin(x)    \                               /                             sin(x)    \ /                              sin(x)            sin(x)           2*cos(x)   \      3*sin(x)          3*cos(x)          3*cos(x)           2*sin(x)          2*sin(x)           3*sin(x)    |
(2 + log(x))      *||cos(x)*log(2 + log(x)) + --------------|  - cos(x)*log(2 + log(x)) - 3*|cos(x)*log(2 + log(x)) + --------------|*|log(2 + log(x))*sin(x) + --------------- + ---------------- - --------------| - -------------- - --------------- - ---------------- + --------------- + ---------------- + ----------------|
                   |\                         x*(2 + log(x))/                               \                         x*(2 + log(x))/ |                          2                 2             2   x*(2 + log(x))|   x*(2 + log(x))    2                 2             2    3                 3             3    3             2|
                   \                                                                                                                  \                         x *(2 + log(x))   x *(2 + log(x))                  /                    x *(2 + log(x))   x *(2 + log(x))    x *(2 + log(x))   x *(2 + log(x))    x *(2 + log(x)) /

$$\left(\log{\left(x \right)} + 2\right)^{\sin{\left(x \right)}} \left(\left(\log{\left(\log{\left(x \right)} + 2 \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x \left(\log{\left(x \right)} + 2\right)}\right)^{3} - 3 \left(\log{\left(\log{\left(x \right)} + 2 \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x \left(\log{\left(x \right)} + 2\right)}\right) \left(\log{\left(\log{\left(x \right)} + 2 \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x \left(\log{\left(x \right)} + 2\right)} + \frac{\sin{\left(x \right)}}{x^{2} \left(\log{\left(x \right)} + 2\right)} + \frac{\sin{\left(x \right)}}{x^{2} \left(\log{\left(x \right)} + 2\right)^{2}}\right) - \log{\left(\log{\left(x \right)} + 2 \right)} \cos{\left(x \right)} - \frac{3 \sin{\left(x \right)}}{x \left(\log{\left(x \right)} + 2\right)} - \frac{3 \cos{\left(x \right)}}{x^{2} \left(\log{\left(x \right)} + 2\right)} - \frac{3 \cos{\left(x \right)}}{x^{2} \left(\log{\left(x \right)} + 2\right)^{2}} + \frac{2 \sin{\left(x \right)}}{x^{3} \left(\log{\left(x \right)} + 2\right)} + \frac{3 \sin{\left(x \right)}}{x^{3} \left(\log{\left(x \right)} + 2\right)^{2}} + \frac{2 \sin{\left(x \right)}}{x^{3} \left(\log{\left(x \right)} + 2\right)^{3}}\right)$$

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