Производная 3^sin(cos(x))

Вы ввели

 sin(cos(x))
3           

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

Подробное решение

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

2. $\frac{d}{d u} 3^{u} = 3^{u} \log{\left (3 \right )}$

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

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

   2. Производная синуса есть косинус:

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

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

      1. Производная косинус есть минус синус:

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

      В результате последовательности правил:

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

   В результате последовательности правил:

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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