Derivada de (x/tgx)+(tgx/x)

Solución

Ha introducido [src]

  x      tan(x)
------ + ------
tan(x)     x

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

x/tan(x) + tan(x)/x

Solución detallada

diferenciamos $\frac{x}{\tan{\left(x \right)}} + \frac{\tan{\left(x \right)}}{x}$ miembro por miembro:
1. 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 $g{\left(x \right)} = \tan{\left(x \right)}$ .
  
  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. 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:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  Ahora aplicamos la regla de la derivada de una divesión:
  
  $\frac{- \frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}}{\tan^{2}{\left(x \right)}}$
2. Se aplica la regla de la derivada parcial:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \tan{\left(x \right)}$ y $g{\left(x \right)} = x$ .
  
  Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
  1. $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
  Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
  1. Según el principio, aplicamos: $x$ tenemos $1$
  Ahora aplicamos la regla de la derivada de una divesión:
  
  $\frac{\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \tan{\left(x \right)}}{x^{2}}$
Como resultado de: $\frac{- \frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}}{\tan^{2}{\left(x \right)}} + \frac{\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \tan{\left(x \right)}}{x^{2}}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

  /             2                                                                                     2                    3                                                                       2\
  |/       2   \                 /       2   \                         /       2   \     /       2   \        /       2   \      /       2   \               2    /       2   \       /       2   \ |
  |\1 + tan (x)/    3*tan(x)   3*\1 + tan (x)/       /       2   \   3*\1 + tan (x)/   3*\1 + tan (x)/    3*x*\1 + tan (x)/    3*\1 + tan (x)/*tan(x)   2*tan (x)*\1 + tan (x)/   5*x*\1 + tan (x)/ |
2*|-------------- - -------- - --------------- - 2*x*\1 + tan (x)/ + --------------- + ---------------- - ------------------ - ---------------------- + ----------------------- + ------------------|
  |      x              4           tan(x)                                   3                3                   4                       2                        x                      2         |
  \                    x                                                    x              tan (x)             tan (x)                   x                                             tan (x)      /

$$2 \left(- \frac{3 x \left(\tan^{2}{\left(x \right)} + 1\right)^{3}}{\tan^{4}{\left(x \right)}} + \frac{5 x \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} - 2 x \left(\tan^{2}{\left(x \right)} + 1\right) + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{3}{\left(x \right)}} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{x} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)}}{x} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x^{2}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{3}} - \frac{3 \tan{\left(x \right)}}{x^{4}}\right)$$

Simplificar

Gráfico

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