Derivada de x*ln(tg(x)*(x/2))

Ha introducido [src]

     /       x\
x*log|tan(x)*-|
     \       2/

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

x*log(tan(x)*(x/2))

Solución detallada

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 $1$
$g{\left(x \right)} = \log{\left(\frac{x}{2} \tan{\left(x \right)} \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. Sustituimos $u = \frac{x}{2} \tan{\left(x \right)}$ .
2. Derivado $\log{\left(u \right)}$ es $\frac{1}{u}$ .
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{x}{2} \tan{\left(x \right)}$ :
  1. Se aplica la regla de la derivada parcial:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = x \tan{\left(x \right)}$ y $g{\left(x \right)} = 2$ .
    
    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 $1$
      $g{\left(x \right)} = \tan{\left(x \right)}$ ; 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 $\frac{d}{d x} f{\left(x \right)}$ :
        
        La derivada del seno es igual al coseno:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
        
        Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
        
        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: $\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}$
    Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
    1. La derivada de una constante $2$ es igual a cero.
    Ahora aplicamos la regla de la derivada de una divesión:
    
    $\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{2 \cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{2}$
  Como resultado de la secuencia de reglas:
  
  $\frac{2 \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{2 \cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{2}\right)}{x \tan{\left(x \right)}}$
Como resultado de: $\frac{2 \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{2 \cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{2}\right)}{\tan{\left(x \right)}} + \log{\left(\frac{x}{2} \tan{\left(x \right)} \right)}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

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Segunda derivada [src]

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

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

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Tercera derivada [src]

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

\frac{\frac{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}}{\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(\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{4 \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{2 \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{4 \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(2 x \left(\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(\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} + \frac{2 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right)}{x^{2}}}{\tan{\left(x \right)}}

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Gráfico

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Expresiones idénticas

Derivada de xln(tg(x)(x/2))

Gráfico:

Definida a trozos:

Solución