Derivada de x*exp(ln(tg(x)))*(ln(tg(x)))(-x)

Ha introducido [src]

   log(tan(x))                 
x*e           *log(tan(x))*(-x)

- x x e^{\log{\left(\tan{\left(x \right)} \right)}} \log{\left(\tan{\left(x \right)} \right)}

((x*exp(log(tan(x))))*log(tan(x)))*(-x)

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = x e^{\log{\left(\tan{\left(x \right)} \right)}} \log{\left(\tan{\left(x \right)} \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x e^{\log{\left(\tan{\left(x \right)} \right)}}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Se aplica la regla de la derivada de una multiplicación:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
    1. Según el principio, aplicamos: $x$ tenemos $1$
    $g{\left(x \right)} = e^{\log{\left(\tan{\left(x \right)} \right)}}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
    1. Sustituimos $u = \log{\left(\tan{\left(x \right)} \right)}$ .
    2. Derivado $e^{u}$ es.
    3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \log{\left(\tan{\left(x \right)} \right)}$ :
      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 $\frac{d}{d x} \tan{\left(x \right)}$ :
        
        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:
        
        $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}$
      Como resultado de la secuencia de reglas:
      
      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    Como resultado de: $\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + e^{\log{\left(\tan{\left(x \right)} \right)}}$
  $g{\left(x \right)} = \log{\left(\tan{\left(x \right)} \right)}$ ; 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: $\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}} + \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + e^{\log{\left(\tan{\left(x \right)} \right)}}\right) \log{\left(\tan{\left(x \right)} \right)}$
$g{\left(x \right)} = - x$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
  1. Según el principio, aplicamos: $x$ tenemos $1$
  Entonces, como resultado: $-1$
Como resultado de: $- x \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}} + \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + e^{\log{\left(\tan{\left(x \right)} \right)}}\right) \log{\left(\tan{\left(x \right)} \right)}\right) - x \log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

Gráfico

Derivada de x*exp(ln(tg(x)))*(ln(tg(x)))(-x)

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Derivada de xexp(ln(tg(x)))(ln(tg(x)))(-x)

Gráfico:

Definida a trozos:

Solución