Производная (x*sin(x))^2

Вы ввели

          2
(x*sin(x)) 

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

Подробное решение

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

2. В силу правила, применим: $u^{2}$ получим $2 u$

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

   1. Применяем правило производной умножения:

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

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

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

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

      1. Производная синуса есть косинус:

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

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

   В результате последовательности правил:

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

4. Теперь упростим:

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

---

Ответ:

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

Первая производная

 2    2                           
x *sin (x)*(2*sin(x) + 2*x*cos(x))
----------------------------------
             x*sin(x)             

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

Вторая производная

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

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

Третья производная

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

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