Производная sin(cos(log(x)))

Вы ввели

sin(cos(log(x)))

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

Подробное решение

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

2. Производная синуса есть косинус:

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

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

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

   2. Производная косинус есть минус синус:

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

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

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

      В результате последовательности правил:

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

   В результате последовательности правил:

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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