Derivada de xsinx/(x+tgx)

Solución

Ha introducido [src]

 x*sin(x) 
----------
x + tan(x)

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

(x*sin(x))/(x + tan(x))

Solución detallada

Se aplica la regla de la derivada parcial:

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

$f{\left(x \right)} = x \sin{\left(x \right)}$ y $g{\left(x \right)} = x + \tan{\left(x \right)}$ .

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 $1$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. La derivada del seno es igual al coseno:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  Como resultado de: $x \cos{\left(x \right)} + \sin{\left(x \right)}$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. diferenciamos $x + \tan{\left(x \right)}$ miembro por miembro:
  1. Según el principio, aplicamos: $x$ tenemos $1$
  2. Reescribimos las funciones para diferenciar:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  3. 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 $\frac{d}{d x} f{\left(x \right)}$ :
    1. La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
    1. 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: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1$
Ahora aplicamos la regla de la derivada de una divesión:

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

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

Respuesta:

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

Gráfica

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Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

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Derivada de xsinx/(x+tgx)

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