Производная exp(sin(tan(x)))

Вы ввели

 sin(tan(x))
e           

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

Подробное решение

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

2. Производная $e^{u}$ само оно.

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

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

   2. Производная синуса есть косинус:

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

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

      1. Есть несколько способов вычислить эту производную.

         Один из способов:

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

      В результате последовательности правил:

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

   В результате последовательности правил:

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

4. Теперь упростим:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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