Производная (log(x)/log(sin(x)))

Вы ввели

   log(x)  
-----------
log(sin(x))

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

Подробное решение

1. Применим правило производной частного:

   $\frac{d}{d x}\left(\frac{f{\left (x \right )}}{g{\left (x \right )}}\right) = \frac{1}{g^{2}{\left (x \right )}} \left(- f{\left (x \right )} \frac{d}{d x} g{\left (x \right )} + g{\left (x \right )} \frac{d}{d x} f{\left (x \right )}\right)$

   $f{\left (x \right )} = \log{\left (x \right )}$ и $g{\left (x \right )} = \log{\left (\sin{\left (x \right )} \right )}$.

   Чтобы найти $\frac{d}{d x} f{\left (x \right )}$:

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

   Чтобы найти $\frac{d}{d x} g{\left (x \right )}$:

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

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

   3. 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:

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

   Теперь применим правило производной деления:

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

2. Теперь упростим:

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

---

Ответ:

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

Первая производная

      1            cos(x)*log(x)   
------------- - -------------------
x*log(sin(x))      2               
                log (sin(x))*sin(x)

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

Вторая производная

                           2                                             2            
  1       log(x)        cos (x)*log(x)           2*cos(x)           2*cos (x)*log(x)  
- -- + ----------- + ------------------- - -------------------- + --------------------
   2   log(sin(x))                  2      x*log(sin(x))*sin(x)      2            2   
  x                  log(sin(x))*sin (x)                          log (sin(x))*sin (x)
--------------------------------------------------------------------------------------
                                     log(sin(x))                                      

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

Третья производная

                            3                      3                                            3                                              2                                               2          
2          3           6*cos (x)*log(x)       6*cos (x)*log(x)       6*cos(x)*log(x)       2*cos (x)*log(x)     2*cos(x)*log(x)           3*cos (x)                3*cos(x)               6*cos (x)       
-- + ------------- - -------------------- - -------------------- - ------------------- - ------------------- - ------------------ + --------------------- + --------------------- + ----------------------
 3   x*log(sin(x))      3            3         2            3         2                                 3      log(sin(x))*sin(x)                    2       2                           2            2   
x                    log (sin(x))*sin (x)   log (sin(x))*sin (x)   log (sin(x))*sin(x)   log(sin(x))*sin (x)                        x*log(sin(x))*sin (x)   x *log(sin(x))*sin(x)   x*log (sin(x))*sin (x)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                               log(sin(x))                                                                                                

$$\frac{1}{\log{\left (\sin{\left (x \right )} \right )}} \left(- \frac{2 \log{\left (x \right )} \cos{\left (x \right )}}{\log{\left (\sin{\left (x \right )} \right )} \sin{\left (x \right )}} - \frac{2 \log{\left (x \right )} \cos^{3}{\left (x \right )}}{\log{\left (\sin{\left (x \right )} \right )} \sin^{3}{\left (x \right )}} - \frac{6 \log{\left (x \right )} \cos{\left (x \right )}}{\log^{2}{\left (\sin{\left (x \right )} \right )} \sin{\left (x \right )}} - \frac{6 \log{\left (x \right )} \cos^{3}{\left (x \right )}}{\log^{2}{\left (\sin{\left (x \right )} \right )} \sin^{3}{\left (x \right )}} - \frac{6 \log{\left (x \right )} \cos^{3}{\left (x \right )}}{\log^{3}{\left (\sin{\left (x \right )} \right )} \sin^{3}{\left (x \right )}} + \frac{3}{x \log{\left (\sin{\left (x \right )} \right )}} + \frac{3 \cos^{2}{\left (x \right )}}{x \log{\left (\sin{\left (x \right )} \right )} \sin^{2}{\left (x \right )}} + \frac{6 \cos^{2}{\left (x \right )}}{x \log^{2}{\left (\sin{\left (x \right )} \right )} \sin^{2}{\left (x \right )}} + \frac{3 \cos{\left (x \right )}}{x^{2} \log{\left (\sin{\left (x \right )} \right )} \sin{\left (x \right )}} + \frac{2}{x^{3}}\right)$$