Производная x^log(x+1)

Вы ввели

 log(x + 1)
x          

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

2. Теперь упростим:

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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