Производная e^(3*x)*sin(sin(sin(x)))

Вы ввели

 3*x                 
E   *sin(sin(sin(x)))

$$e^{3 x} \sin{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )}$$

Подробное решение

1. Применяем правило производной умножения:

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

   $f{\left (x \right )} = e^{3 x}$; найдём $\frac{d}{d x} f{\left (x \right )}$:

   1. Заменим $u = 3 x$.

   2. Производная $e^{u}$ само оно.

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

      1. Производная произведения константы на функцию есть произведение этой константы на производную данной функции.

         1. В силу правила, применим: $x$ получим $1$

         Таким образом, в результате: $3$

      В результате последовательности правил:

      $3 e^{3 x}$

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

   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:

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

   В результате: $3 e^{3 x} \sin{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )} + e^{3 x} \cos{\left (x \right )} \cos{\left (\sin{\left (x \right )} \right )} \cos{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )}$

2. Теперь упростим:

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

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Ответ:

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

Первая производная

   3*x                                                         3*x
3*e   *sin(sin(sin(x))) + cos(x)*cos(sin(x))*cos(sin(sin(x)))*e   

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

Упростить

Вторая производная

/                        2       2                               2                                                                                                              \  3*x
\9*sin(sin(sin(x))) - 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) + 6*cos(x)*cos(sin(x))*cos(sin(sin(x)))/*e   

$$\left(- \sin{\left (x \right )} \cos{\left (\sin{\left (x \right )} \right )} \cos{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )} - \sin{\left (\sin{\left (x \right )} \right )} \cos^{2}{\left (x \right )} \cos{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )} - \sin{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )} \cos^{2}{\left (x \right )} \cos^{2}{\left (\sin{\left (x \right )} \right )} + 9 \sin{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )} + 6 \cos{\left (x \right )} \cos{\left (\sin{\left (x \right )} \right )} \cos{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )}\right) e^{3 x}$$

Упростить

Третья производная

/                         3       3                               3                                        2       2                                 2                                                                                                                         3                                                    2                                                                                      \  3*x
\27*sin(sin(sin(x))) - cos (x)*cos (sin(x))*cos(sin(sin(x))) - cos (x)*cos(sin(x))*cos(sin(sin(x))) - 9*cos (x)*cos (sin(x))*sin(sin(sin(x))) - 9*cos (x)*cos(sin(sin(x)))*sin(sin(x)) - 9*cos(sin(x))*cos(sin(sin(x)))*sin(x) + 26*cos(x)*cos(sin(x))*cos(sin(sin(x))) + 3*cos (x)*cos(sin(x))*sin(sin(x))*sin(sin(sin(x))) + 3*cos (sin(x))*cos(x)*sin(x)*sin(sin(sin(x))) + 3*cos(x)*cos(sin(sin(x)))*sin(x)*sin(sin(x))/*e   

$$\left(3 \sin{\left (x \right )} \sin{\left (\sin{\left (x \right )} \right )} \cos{\left (x \right )} \cos{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )} + 3 \sin{\left (x \right )} \sin{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )} \cos{\left (x \right )} \cos^{2}{\left (\sin{\left (x \right )} \right )} - 9 \sin{\left (x \right )} \cos{\left (\sin{\left (x \right )} \right )} \cos{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )} + 3 \sin{\left (\sin{\left (x \right )} \right )} \sin{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )} \cos^{3}{\left (x \right )} \cos{\left (\sin{\left (x \right )} \right )} - 9 \sin{\left (\sin{\left (x \right )} \right )} \cos^{2}{\left (x \right )} \cos{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )} - 9 \sin{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )} \cos^{2}{\left (x \right )} \cos^{2}{\left (\sin{\left (x \right )} \right )} + 27 \sin{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )} - \cos^{3}{\left (x \right )} \cos^{3}{\left (\sin{\left (x \right )} \right )} \cos{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )} - \cos^{3}{\left (x \right )} \cos{\left (\sin{\left (x \right )} \right )} \cos{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )} + 26 \cos{\left (x \right )} \cos{\left (\sin{\left (x \right )} \right )} \cos{\left (\sin{\left (\sin{\left (x \right )} \right )} \right )}\right) e^{3 x}$$

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