Производная sin(cos(tan(x)))

Вы ввели

sin(cos(tan(x)))

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

d                   
--(sin(cos(tan(x))))
dx                  

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

Подробное решение

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

2. Производная синуса есть косинус:

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

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

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

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

      1. Перепишем функции, чтобы дифференцировать:

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

      2. Применим правило производной частного:

         $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

         $f{\left(x \right)} = \sin{\left(x \right)}$ и $g{\left(x \right)} = \cos{\left(x \right)}$.

         Чтобы найти $\frac{d}{d x} f{\left(x \right)}$:

         1. Производная синуса есть косинус:

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

         Чтобы найти $\frac{d}{d x} g{\left(x \right)}$:

         1. Производная косинус есть минус синус:

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

         Теперь применим правило производной деления:

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

      В результате последовательности правил:

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

   В результате последовательности правил:

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

4. Теперь упростим:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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

График