Производная 1/(x*sin(x))

Вы ввели

     1    
1*--------
  x*sin(x)

$$1 \cdot \frac{1}{x \sin{\left(x \right)}}$$

d /     1    \
--|1*--------|
dx\  x*sin(x)/

$$\frac{d}{d x} 1 \cdot \frac{1}{x \sin{\left(x \right)}}$$

Подробное решение

1. Применим правило производной частного:

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

   $f{\left(x \right)} = 1$ и $g{\left(x \right)} = x \sin{\left(x \right)}$.

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

   1. Производная постоянной $1$ равна нулю.

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

   1. Применяем правило производной умножения:

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

      $f{\left(x \right)} = x$; 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: $x \cos{\left(x \right)} + \sin{\left(x \right)}$

   Теперь применим правило производной деления:

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

2. Теперь упростим:

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

---

Ответ:

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

Первая производная

   1                         
--------*(-sin(x) - x*cos(x))
x*sin(x)                     
-----------------------------
           x*sin(x)          

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

Вторая производная

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

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

Третья производная

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

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

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