Derivada de x/(ln(tan(x)))

Solución

Ha introducido [src]

     x     
-----------
log(tan(x))

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

x/log(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 $g{\left(x \right)} = \log{\left(\tan{\left(x \right)} \right)}$ .

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. Sustituimos $u = \tan{\left(x \right)}$ .
2. Derivado $\log{\left(u \right)}$ es $\frac{1}{u}$ .
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \tan{\left(x \right)}$ :
  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 la secuencia de reglas:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}$
Ahora aplicamos la regla de la derivada de una divesión:

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

Gráfico

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Derivada de x/(ln(tan(x)))

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