Derivada de x*sinx/(1+tgx)

Solución

Ha introducido [src]

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

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

(x*sin(x))/(1 + 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 $g{\left(x \right)} = \tan{\left(x \right)} + 1$ .

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 $\tan{\left(x \right)} + 1$ miembro por miembro:
  1. La derivada de una constante $1$ es igual a cero.
  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: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Ahora aplicamos la regla de la derivada de una divesión:

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

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Gráfico

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Derivada de x*sinx/(1+tgx)

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