Derivada de (x*lnctgx)/sinx

Solución

Ha introducido [src]

x*log(cot(x))
-------------
    sin(x)

$$\frac{x \log{\left(\cot{\left(x \right)} \right)}}{\sin{\left(x \right)}}$$

(x*log(cot(x)))/sin(x)

Solución detallada

Se aplica la regla de la derivada parcial:

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

$f{\left(x \right)} = x \log{\left(\cot{\left(x \right)} \right)}$ y $g{\left(x \right)} = \sin{\left(x \right)}$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Según el principio, aplicamos: $x$ tenemos $1$
  $g{\left(x \right)} = \log{\left(\cot{\left(x \right)} \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Sustituimos $u = \cot{\left(x \right)}$ .
  2. Derivado $\log{\left(u \right)}$ es $\frac{1}{u}$ .
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \cot{\left(x \right)}$ :
    1. Hay varias formas de calcular esta derivada.
      Method #1
      
      Reescribimos las funciones para diferenciar:
      
      $\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}$
      
      Sustituimos $u = \tan{\left(x \right)}$ .
      
      Según el principio, aplicamos: $\frac{1}{u}$ tenemos $- \frac{1}{u^{2}}$
      
      Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \tan{\left(x \right)}$ :
      
      Reescribimos las funciones para diferenciar:
      
      $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
      
      Se aplica la regla de la derivada parcial:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(x \right)}$ y $g{\left(x \right)} = \cos{\left(x \right)}$ .
      
      Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
      
      La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      
      Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
      
      La derivada del coseno es igual a menos el seno:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      
      Ahora aplicamos la regla de la derivada de una divesión:
      
      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
      
      Como resultado de la secuencia de reglas:
      
      $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
      Method #2
      
      Reescribimos las funciones para diferenciar:
      
      $\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
      
      Se aplica la regla de la derivada parcial:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(x \right)}$ y $g{\left(x \right)} = \sin{\left(x \right)}$ .
      
      Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
      
      La derivada del coseno es igual a menos el seno:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      
      Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
      
      La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      
      Ahora aplicamos la regla de la derivada de una divesión:
      
      $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
    Como resultado de la secuencia de reglas:
    
    $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)} \cot{\left(x \right)}}$
  Como resultado de: $- \frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)} \cot{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)}$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. La derivada del seno es igual al coseno:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
Ahora aplicamos la regla de la derivada de una divesión:

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

 /                                                                                                                                                             /  /                             2\                  \                                              \ 
 |                                                                                                                                                             |  |                /       2   \ |     /       2   \|            /         2   \                   | 
 |                                                                                                                                                             |  |         2      \1 + cot (x)/ |   2*\1 + cot (x)/|            |    6*cos (x)|                   | 
 |                                2                                                                          /                        2                  \   3*|x*|2 + 2*cot (x) - --------------| - ---------------|*cos(x)   x*|5 + ---------|*cos(x)*log(cot(x))| 
 |                   /       2   \      /         2   \ /                 /       2   \\                     |           /       2   \      /       2   \|     |  |                      2       |        cot(x)    |            |        2    |                   | 
 |          2      3*\1 + cot (x)/      |    2*cos (x)| |               x*\1 + cot (x)/|       /       2   \ |           \1 + cot (x)/    2*\1 + cot (x)/|     \  \                   cot (x)    /                  /            \     sin (x) /                   | 
-|-6 - 6*cot (x) + ---------------- + 3*|1 + ---------|*|-log(cot(x)) + ---------------| + 2*x*\1 + cot (x)/*|2*cot(x) + -------------- - ---------------| + --------------------------------------------------------------- + ------------------------------------| 
 |                        2             |        2    | \                    cot(x)    /                     |                 3               cot(x)    |                                sin(x)                                              sin(x)               | 
 \                     cot (x)          \     sin (x) /                                                      \              cot (x)                      /                                                                                                         / 
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                sin(x)

$$- \frac{\frac{x \left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(\cot{\left(x \right)} \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2 x \left(\cot^{2}{\left(x \right)} + 1\right) \left(\frac{\left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{3}{\left(x \right)}} - \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot{\left(x \right)}} + 2 \cot{\left(x \right)}\right) + 3 \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \left(\frac{x \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot{\left(x \right)}} - \log{\left(\cot{\left(x \right)} \right)}\right) + \frac{3 \left(x \left(- \frac{\left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{2}{\left(x \right)}} + 2 \cot^{2}{\left(x \right)} + 2\right) - \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{3 \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{2}{\left(x \right)}} - 6 \cot^{2}{\left(x \right)} - 6}{\sin{\left(x \right)}}$$

Simplificar

Gráfico

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Expresiones idénticas

Derivada de (x*lnctgx)/sinx

Gráfico:

Definida a trozos:

Solución

Method #1

Method #2