Производная tan(sin(cos(x)))

Вы ввели

tan(sin(cos(x)))

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

Подробное решение

1. Есть несколько способов вычислить эту производную.

   Один из способов:

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

   2. $\frac{d}{d u} \tan{\left (u \right )} = \frac{1}{\cos^{2}{\left (u \right )}}$

   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:

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

2. Теперь упростим:

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

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Ответ:

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

Первая производная

 /       2             \                   
-\1 + tan (sin(cos(x)))/*cos(cos(x))*sin(x)

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

Вторая производная

/       2             \ /     2                                            2            2                    \
\1 + tan (sin(cos(x)))/*\- sin (x)*sin(cos(x)) - cos(x)*cos(cos(x)) + 2*cos (cos(x))*sin (x)*tan(sin(cos(x)))/

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

Третья производная

/       2             \ /   2                                              3            2       2                     3            2    /       2             \        2                                        2                                                          \       
\1 + tan (sin(cos(x)))/*\sin (x)*cos(cos(x)) - 3*cos(x)*sin(cos(x)) - 4*cos (cos(x))*sin (x)*tan (sin(cos(x))) - 2*cos (cos(x))*sin (x)*\1 + tan (sin(cos(x)))/ + 6*cos (cos(x))*cos(x)*tan(sin(cos(x))) + 6*sin (x)*cos(cos(x))*sin(cos(x))*tan(sin(cos(x))) + cos(cos(x))/*sin(x)

$$\left(\tan^{2}{\left (\sin{\left (\cos{\left (x \right )} \right )} \right )} + 1\right) \left(- 2 \left(\tan^{2}{\left (\sin{\left (\cos{\left (x \right )} \right )} \right )} + 1\right) \sin^{2}{\left (x \right )} \cos^{3}{\left (\cos{\left (x \right )} \right )} + 6 \sin^{2}{\left (x \right )} \sin{\left (\cos{\left (x \right )} \right )} \cos{\left (\cos{\left (x \right )} \right )} \tan{\left (\sin{\left (\cos{\left (x \right )} \right )} \right )} - 4 \sin^{2}{\left (x \right )} \cos^{3}{\left (\cos{\left (x \right )} \right )} \tan^{2}{\left (\sin{\left (\cos{\left (x \right )} \right )} \right )} + \sin^{2}{\left (x \right )} \cos{\left (\cos{\left (x \right )} \right )} - 3 \sin{\left (\cos{\left (x \right )} \right )} \cos{\left (x \right )} + 6 \cos{\left (x \right )} \cos^{2}{\left (\cos{\left (x \right )} \right )} \tan{\left (\sin{\left (\cos{\left (x \right )} \right )} \right )} + \cos{\left (\cos{\left (x \right )} \right )}\right) \sin{\left (x \right )}$$