Производная cos(sin(x))*cos(x)

Вы ввели

cos(sin(x))*cos(x)

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

Подробное решение

1. Применяем правило производной умножения:

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

   $f{\left (x \right )} = \cos{\left (\sin{\left (x \right )} \right )}$; найдём $\frac{d}{d x} f{\left (x \right )}$:

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

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

      1. Производная синуса есть косинус:

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

      В результате последовательности правил:

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

   $g{\left (x \right )} = \cos{\left (x \right )}$; найдём $\frac{d}{d x} g{\left (x \right )}$:

   1. Производная косинус есть минус синус:

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

   В результате: $- \sin{\left (x \right )} \cos{\left (\sin{\left (x \right )} \right )} - \sin{\left (\sin{\left (x \right )} \right )} \cos^{2}{\left (x \right )}$

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Ответ:

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

Первая производная

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

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

Вторая производная

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

$$\left(3 \sin{\left (x \right )} \sin{\left (\sin{\left (x \right )} \right )} - \cos^{2}{\left (x \right )} \cos{\left (\sin{\left (x \right )} \right )} - \cos{\left (\sin{\left (x \right )} \right )}\right) \cos{\left (x \right )}$$

Третья производная

   4                                            2                       2                       2                      
cos (x)*sin(sin(x)) + cos(sin(x))*sin(x) - 3*sin (x)*sin(sin(x)) + 4*cos (x)*sin(sin(x)) + 6*cos (x)*cos(sin(x))*sin(x)

$$- 3 \sin^{2}{\left (x \right )} \sin{\left (\sin{\left (x \right )} \right )} + 6 \sin{\left (x \right )} \cos^{2}{\left (x \right )} \cos{\left (\sin{\left (x \right )} \right )} + \sin{\left (x \right )} \cos{\left (\sin{\left (x \right )} \right )} + \sin{\left (\sin{\left (x \right )} \right )} \cos^{4}{\left (x \right )} + 4 \sin{\left (\sin{\left (x \right )} \right )} \cos^{2}{\left (x \right )}$$