Производная sin(sin(sin(x)))

Вы ввели

sin(sin(sin(x)))

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

d                   
--(sin(sin(sin(x))))
dx                  

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

Подробное решение

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

2. Производная синуса есть косинус:

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

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

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

   2. Производная синуса есть косинус:

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

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

      1. Производная синуса есть косинус:

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

      В результате последовательности правил:

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

   В результате последовательности правил:

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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

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