Derivada de x/(x^2*tgx)

Ha introducido [src]

    x    
---------
 2       
x *tan(x)

\frac{x}{x^{2} \tan{\left(x \right)}}

x/((x^2*tan(x)))

Solución detallada

Se aplica la regla de la derivada parcial:

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

$f{\left(x \right)} = x$ y $g{\left(x \right)} = x^{2} \tan{\left(x \right)}$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. Según el principio, aplicamos: $x$ tenemos $1$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{2}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Según el principio, aplicamos: $x^{2}$ tenemos $2 x$
  $g{\left(x \right)} = \tan{\left(x \right)}$ ; 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: $\frac{x^{2} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + 2 x \tan{\left(x \right)}$
Ahora aplicamos la regla de la derivada de una divesión:

$\frac{x^{2} \tan{\left(x \right)} - x \left(\frac{x^{2} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + 2 x \tan{\left(x \right)}\right)}{x^{4} \tan^{2}{\left(x \right)}}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

               2 /       2   \             
    1       - x *\1 + tan (x)/ - 2*x*tan(x)
--------- + -------------------------------
 2                      3    2             
x *tan(x)              x *tan (x)

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

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Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

                                                                                                                                                                                                                                                                             //           2   \                                  /    /       2   \    2 /       2   \                \     /             /       2   \\   /       2   \ /             /       2   \\\                                                                                                                 /           2   \                                  /           2   \                                                                        /           2   \                                                                                                                                                     
                                      /                                2                                                        \                                  /                                 2                  \                                                    ||2   1 + tan (x)| /             /       2   \\   2*\2*x*\1 + tan (x)/ + x *\1 + tan (x)/*tan(x) + tan(x)/   2*\2*tan(x) + x*\1 + tan (x)//   \1 + tan (x)/*\2*tan(x) + x*\1 + tan (x)//|                                                                              2                                  |2   1 + tan (x)| /             /       2   \\     |2   1 + tan (x)| /    /       2   \    2 /       2   \                \   /       2   \ |2   1 + tan (x)| /             /       2   \\                                                                                                                        
     /             /       2   \\     |         2       2 /       2   \       2    2    /       2   \       /       2   \       |                                  |                    /       2   \      /       2   \|                                                  3*||- + -----------|*\2*tan(x) + x*\1 + tan (x)// - -------------------------------------------------------- + ------------------------------ + ------------------------------------------|      /    /       2   \    2 /       2   \                \     /       2   \  /             /       2   \\   2*|- + -----------|*\2*tan(x) + x*\1 + tan (x)//   2*|- + -----------|*\2*x*\1 + tan (x)/ + x *\1 + tan (x)/*tan(x) + tan(x)/   \1 + tan (x)/*|- + -----------|*\2*tan(x) + x*\1 + tan (x)//     /       2   \ /             /       2   \\     /       2   \ /    /       2   \    2 /       2   \                \
  10*\2*tan(x) + x*\1 + tan (x)//   2*\3 + 3*tan (x) + x *\1 + tan (x)/  + 2*x *tan (x)*\1 + tan (x)/ + 6*x*\1 + tan (x)/*tan(x)/     /             /       2   \\ |        2      3    \1 + tan (x)/    2*\1 + tan (x)/|     /       2   \ /             /       2   \\     \\x      tan(x)  /                                                           x                                             x                                    tan(x)                  /   12*\2*x*\1 + tan (x)/ + x *\1 + tan (x)/*tan(x) + tan(x)/   3*\1 + tan (x)/ *\2*tan(x) + x*\1 + tan (x)//     \x      tan(x)  /                                  \x      tan(x)  /                                                                        \x      tan(x)  /                                8*\1 + tan (x)/*\2*tan(x) + x*\1 + tan (x)//   6*\1 + tan (x)/*\2*x*\1 + tan (x)/ + x *\1 + tan (x)/*tan(x) + tan(x)/
- ------------------------------- - --------------------------------------------------------------------------------------------- - 2*\2*tan(x) + x*\1 + tan (x)//*|-1 - tan (x) + -- + -------------- + ---------------| + 2*\1 + tan (x)/*\2*tan(x) + x*\1 + tan (x)// + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------- - --------------------------------------------- - ------------------------------------------------ + -------------------------------------------------------------------------- - ------------------------------------------------------------ - -------------------------------------------- + ----------------------------------------------------------------------
                  2                                                               x                                                                                |                2         2              x*tan(x)   |                                                                                                                                               x                                                                                                                             2                                                    2                                                x                                                               x                                                                   tan(x)                                                x*tan(x)                                                    x*tan(x)                               
                 x                                                                                                                                                 \               x       tan (x)                      /                                                                                                                                                                                                                                                                            x                                                  tan (x)                                                                                                                                                                                                                                                                                                                                          
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                                                                                                                                                                                                                                                                                                                                                                                                                                                 2    2                                                                                                                                                                                                                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                                                                                                                                                                                                                x *tan (x)

\frac{- \frac{\left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \tan{\left(x \right)}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}} + \frac{2}{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) + 2 \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) + 2 \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) - 2 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \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{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x \tan{\left(x \right)}} + \frac{3}{x^{2}}\right) - \frac{2 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \tan{\left(x \right)}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}} + \frac{2}{x}\right)}{x} - \frac{8 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{x \tan{\left(x \right)}} + \frac{2 \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}} + \frac{2}{x}\right) \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right)}{x} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right)}{x \tan{\left(x \right)}} + \frac{3 \left(\left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \tan{\left(x \right)}\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}} + \frac{2}{x}\right) + \frac{\left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + \frac{2 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \tan{\left(x \right)}\right)}{x} - \frac{2 \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right)}{x}\right)}{x} - \frac{2 \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 6 x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 3 \tan^{2}{\left(x \right)} + 3\right)}{x} - \frac{10 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \tan{\left(x \right)}\right)}{x^{2}} + \frac{12 \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right)}{x^{2}}}{x^{2} \tan^{2}{\left(x \right)}}

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Gráfico

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Derivada de x/(x^2*tgx)

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