Derivada de sin(x)/(1+tan(x))

Solución

Ha introducido [src]

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

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

sin(x)/(1 + tan(x))

Solución detallada

Se aplica la regla de la derivada parcial:

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

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

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. 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. 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:

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

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

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

Simplificar

Gráfico

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Derivada de sin(x)/(1+tan(x))

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