Производная (x+sin(x))^(log(x))

Вы ввели [src]

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

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

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

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

Преобразовать

Подробное решение

Не могу найти шаги в поиске этой производной.

Но производная

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

Ответ:

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

График

Построить график f(x)

Построить график f'(x)

Первая производная [src]

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

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

Упростить

Вторая производная [src]

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

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

Упростить

Третья производная [src]

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

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

Упростить

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Производная (x+sin(x))^(log(x))

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