Производная log(x)/log(sin(x))

Вы ввели

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

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

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

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

Подробное решение

1. Применим правило производной частного:

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

   $f{\left(x \right)} = \log{\left(x \right)}$ 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:

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

2. Теперь упростим:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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

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