Derivada de (x-sinx)/(xtanx)

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

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

               /                                                                                                                                                                                                                                                                                                                                     /           2   \                                                                                                                                                   /           2   \                                                                        \                                                                                                                                                                                                                                 
               |                                                                                                                                                                                                                                                 /                                 2              \                                  |1   1 + tan (x)| /  /       2   \         \                                                                           2                              /       2   \ |1   1 + tan (x)| /  /       2   \         \                                             |                                                                                                                                                                                                                                 
               |    /       2        /       2   \       \                                                  /           2   \                                                                                                                                    |                    /       2   \           2   |     /  /       2   \         \   |- + -----------|*\x*\1 + tan (x)/ + tan(x)/     /       2   \ /       2        /       2   \       \     /       2   \  /  /       2   \         \   \1 + tan (x)/*|- + -----------|*\x*\1 + tan (x)/ + tan(x)/     /       2   \ /  /       2   \         \|                                                                                                                                                                                                                                 
               |  6*\1 + tan (x) + x*\1 + tan (x)/*tan(x)/     /       2   \ /  /       2   \         \     |1   1 + tan (x)| /       2        /       2   \       \     /       2   \ /             /       2   \          2   \     /  /       2   \         \ |     1       2      \1 + tan (x)/    1 + tan (x)|   3*\x*\1 + tan (x)/ + tan(x)/   \x      tan(x)  /                              6*\1 + tan (x)/*\1 + tan (x) + x*\1 + tan (x)/*tan(x)/   3*\1 + tan (x)/ *\x*\1 + tan (x)/ + tan(x)/                 \x      tan(x)  /                              4*\1 + tan (x)/*\x*\1 + tan (x)/ + tan(x)/|                                                         /                  /       2   \            /           2   \                              /       2   \ /  /       2   \         \                           \         
  (x - sin(x))*|- ---------------------------------------- - 2*\1 + tan (x)/*\x*\1 + tan (x)/ + tan(x)/ - 2*|- + -----------|*\1 + tan (x) + x*\1 + tan (x)/*tan(x)/ + 2*\1 + tan (x)/*\3*tan(x) + x*\1 + tan (x)/ + 2*x*tan (x)/ + 2*\x*\1 + tan (x)/ + tan(x)/*|-1 + -- - tan (x) + -------------- + -----------| + ---------------------------- + -------------------------------------------- - ------------------------------------------------------ + ------------------------------------------- + ---------------------------------------------------------- + ------------------------------------------|                                                         |         2      x*\1 + tan (x)/ + tan(x)   |1   1 + tan (x)| /  /       2   \         \   \1 + tan (x)/*\x*\1 + tan (x)/ + tan(x)/       /       2   \       |         
               |                     x                                                                      \x      tan(x)  /                                                                                                                                    |      2                   2            x*tan(x) |                 2                                     x                                                 tan(x)                                                2                                                  tan(x)                                              x*tan(x)                 |     /  /       2   \         \          3*(-1 + cos(x))*|2 + 2*tan (x) - ------------------------ - |- + -----------|*\x*\1 + tan (x)/ + tan(x)/ - ---------------------------------------- + 2*x*\1 + tan (x)/*tan(x)|         
               \                                                                                                                                                                                                                                                 \     x                 tan (x)                  /                x                                                                                                                                           tan (x)                                                                                                                            /   3*\x*\1 + tan (x)/ + tan(x)/*sin(x)                   \                           x               \x      tan(x)  /                                               tan(x)                                            /         
- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + cos(x)
                                                                                                                                                                                                                                                                                                       x*tan(x)                                                                                                                                                                                                                                                                                                                     x*tan(x)                                                                                                    x*tan(x)                                                                                            
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                      x*tan(x)

$$\frac{\cos{\left(x \right)} - \frac{\left(x - \sin{\left(x \right)}\right) \left(\frac{\left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}} + \frac{1}{x}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + \frac{3 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} - 2 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} - \tan^{2}{\left(x \right)} - 1 + \frac{\tan^{2}{\left(x \right)} + 1}{x \tan{\left(x \right)}} + \frac{1}{x^{2}}\right) - 2 \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}} + \frac{1}{x}\right) \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right) + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 x \tan^{2}{\left(x \right)} + 3 \tan{\left(x \right)}\right) - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + \frac{\left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}} + \frac{1}{x}\right)}{x} + \frac{4 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{x \tan{\left(x \right)}} - \frac{6 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{3 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right)}{x^{2}}\right)}{x \tan{\left(x \right)}} - \frac{3 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \sin{\left(x \right)}}{x \tan{\left(x \right)}} + \frac{3 \left(\cos{\left(x \right)} - 1\right) \left(2 x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}} + \frac{1}{x}\right) - \frac{\left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + 2 \tan^{2}{\left(x \right)} + 2 - \frac{x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}}{x}\right)}{x \tan{\left(x \right)}}}{x \tan{\left(x \right)}}$$

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Derivada de (x-sinx)/(xtanx)

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