Производная (1+log(sin(x)))^n

Вы ввели

                 n
(1 + log(sin(x))) 

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

Подробное решение

1. Заменим $u = \log{\left (\sin{\left (x \right )} \right )} + 1$.

2. В силу правила, применим: $u^{n}$ получим $\frac{n u^{n}}{u}$

3. Затем примените цепочку правил. Умножим на $\frac{d}{d x}\left(\log{\left (\sin{\left (x \right )} \right )} + 1\right)$:

   1. дифференцируем $\log{\left (\sin{\left (x \right )} \right )} + 1$ почленно:

      1. Производная постоянной $1$ равна нулю.

      2. Заменим $u = \sin{\left (x \right )}$.

      3. Производная $\log{\left (u \right )}$ является $\frac{1}{u}$.

      4. Затем примените цепочку правил. Умножим на $\frac{d}{d x} \sin{\left (x \right )}$:

         1. Производная синуса есть косинус:

            $\frac{d}{d x} \sin{\left (x \right )} = \cos{\left (x \right )}$

         В результате последовательности правил:

         $\frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}$

      В результате: $\frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}$

   В результате последовательности правил:

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

4. Теперь упростим:

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

---

Ответ:

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

Первая производная

                   n       
n*(1 + log(sin(x))) *cos(x)
---------------------------
  (1 + log(sin(x)))*sin(x) 

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

Вторая производная

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

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

Третья производная

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

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