Derivada de y=lnlntgx

Solución

Ha introducido [src]

log(x)*log(tan(x))

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

log(x)*log(tan(x))

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = \log{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Derivado $\log{\left(x \right)}$ es $\frac{1}{x}$ .
$g{\left(x \right)} = \log{\left(\tan{\left(x \right)} \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
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:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}$
Como resultado de: $\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}} + \frac{\log{\left(\tan{\left(x \right)} \right)}}{x}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

Gráfico

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