Derivada de y=ctgx*ln(tg)x

Solución

Ha introducido [src]

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

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

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

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = \log{\left(\tan{\left(x \right)} \right)} \cot{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \cot{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Hay varias formas de calcular esta derivada.
    Method #1
    1. Reescribimos las funciones para diferenciar:
      
      $\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}$
    2. Sustituimos $u = \tan{\left(x \right)}$ .
    3. Según el principio, aplicamos: $\frac{1}{u}$ tenemos $- \frac{1}{u^{2}}$
    4. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \tan{\left(x \right)}$ :
      
      Reescribimos las funciones para diferenciar:
      
      $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
      
      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 la secuencia de reglas:
      
      $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
    Method #2
    1. Reescribimos las funciones para diferenciar:
      
      $\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
    2. Se aplica la regla de la derivada parcial:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(x \right)}$ y $g{\left(x \right)} = \sin{\left(x \right)}$ .
      
      Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
      
      La derivada del coseno es igual a menos el seno:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      
      Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
      
      La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      
      Ahora aplicamos la regla de la derivada de una divesión:
      
      $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
  $g{\left(x \right)} = \log{\left(\tan{\left(x \right)} \right)}$ ; 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. $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
    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: $- \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(\tan{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \cot{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}$
$g{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. Según el principio, aplicamos: $x$ tenemos $1$
Como resultado de: $x \left(- \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(\tan{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \cot{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}\right) + \log{\left(\tan{\left(x \right)} \right)} \cot{\left(x \right)}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

Gráfico

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Derivada de y=ctgx*ln(tg)x

Gráfico:

Definida a trozos:

Solución

Method #1

Method #2