Derivada de y=2^tgx^e

Solución

Ha introducido [src]

    E   
 tan (x)
2

$$2^{\tan^{e}{\left(x \right)}}$$

2^(tan(x)^E)

Solución detallada

Sustituimos $u = \tan^{e}{\left(x \right)}$ .
$\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \tan^{e}{\left(x \right)}$ :
1. Sustituimos $u = \tan{\left(x \right)}$ .
2. Según el principio, aplicamos: $u^{e}$ tenemos $\frac{e u^{e}}{u}$
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:
  
  $\frac{e \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{e}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}$
Como resultado de la secuencia de reglas:

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

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

Respuesta:

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

Gráfica

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Primera derivada [src]

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

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

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Segunda derivada [src]

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

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

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Tercera derivada [src]

                                 /                                            2                2                       2                                    2                                         2                                 2                                                     \       
      E                          |             /       2   \     /       2   \    /       2   \   2       /       2   \        /       2   \   /       2   \     2       2*E     2       /       2   \     E               /       2   \     E     2                 E    /       2   \       |       
   tan (x)    E    /       2   \ |           4*\1 + tan (x)/   2*\1 + tan (x)/    \1 + tan (x)/ *e    3*E*\1 + tan (x)/    6*E*\1 + tan (x)/   \1 + tan (x)/ *log (2)*tan   (x)*e    3*E*\1 + tan (x)/ *tan (x)*log(2)   3*\1 + tan (x)/ *tan (x)*e *log(2)   6*E*tan (x)*\1 + tan (x)/*log(2)|       
E*2       *tan (x)*\1 + tan (x)/*|4*tan(x) - --------------- + ---------------- + ----------------- - ------------------ + ----------------- + ----------------------------------- - --------------------------------- + ---------------------------------- + --------------------------------|*log(2)
                                 |                tan(x)              3                   3                   3                  tan(x)                         3                                    3                                   3                                 tan(x)             |       
                                 \                                 tan (x)             tan (x)             tan (x)                                           tan (x)                              tan (x)                             tan (x)                                                 /

$$2^{\tan^{e}{\left(x \right)}} e \left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2} e^{2} \log{\left(2 \right)}^{2} \tan^{2 e}{\left(x \right)}}{\tan^{3}{\left(x \right)}} - \frac{3 e \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(2 \right)} \tan^{e}{\left(x \right)}}{\tan^{3}{\left(x \right)}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} e^{2} \log{\left(2 \right)} \tan^{e}{\left(x \right)}}{\tan^{3}{\left(x \right)}} - \frac{3 e \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{3}{\left(x \right)}} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{3}{\left(x \right)}} + \frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2} e^{2}}{\tan^{3}{\left(x \right)}} + \frac{6 e \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(2 \right)} \tan^{e}{\left(x \right)}}{\tan{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + \frac{6 e \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + 4 \tan{\left(x \right)}\right) \log{\left(2 \right)} \tan^{e}{\left(x \right)}$$

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Gráfico

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Derivada de y=2^tgx^e

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Solución