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y=2^tgx^e
Derivada de la función/ y=2^tgx/ y=2^tgx^e

Derivada de y=2^tgx^e

Función f() - derivada -er orden en el punto
v

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src] 
    E   
 tan (x)
2
2tan⁡e(x)2^{\tan^{e}{\left(x \right)}} 
2^(tan(x)^E)
Solución detallada

  1. Sustituimos u=tan⁡e(x)u = \tan^{e}{\left(x \right)}.

  2. ddu2u=2ulog⁡(2)\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddxtan⁡e(x)\frac{d}{d x} \tan^{e}{\left(x \right)}:

    1. Sustituimos u=tan⁡(x)u = \tan{\left(x \right)}.

    2. Según el principio, aplicamos: ueu^{e} tenemos eueu\frac{e u^{e}}{u}

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddxtan⁡(x)\frac{d}{d x} \tan{\left(x \right)}:

      1. Reescribimos las funciones para diferenciar:

        tan⁡(x)=sin⁡(x)cos⁡(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

      2. Se aplica la regla de la derivada parcial:

        ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} y g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}.

        Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. La derivada del seno es igual al coseno:

          ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

        Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. La derivada del coseno es igual a menos el seno:

          ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

        Ahora aplicamos la regla de la derivada de una divesión:

        sin⁡2(x)+cos⁡2(x)cos⁡2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

      Como resultado de la secuencia de reglas:

      e(sin⁡2(x)+cos⁡2(x))tan⁡e(x)cos⁡2(x)tan⁡(x)\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:

    2tan⁡e(x)e(sin⁡2(x)+cos⁡2(x))log⁡(2)tan⁡e(x)cos⁡2(x)tan⁡(x)\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)}}

  4. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-101001e87
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
      E                                
   tan (x)    E    /       2   \       
E*2       *tan (x)*\1 + tan (x)/*log(2)
---------------------------------------
                 tan(x)
2tan⁡e(x)e(tan⁡2(x)+1)log⁡(2)tan⁡e(x)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)}}
Simplificar
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)            /
2tan⁡e(x)e(tan⁡2(x)+1)(e(tan⁡2(x)+1)log⁡(2)tan⁡e(x)tan⁡2(x)−tan⁡2(x)+1tan⁡2(x)+e(tan⁡2(x)+1)tan⁡2(x)+2)log⁡(2)tan⁡e(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)}
Simplificar
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)                                                 /
2tan⁡e(x)e(tan⁡2(x)+1)((tan⁡2(x)+1)2e2log⁡(2)2tan⁡2e(x)tan⁡3(x)−3e(tan⁡2(x)+1)2log⁡(2)tan⁡e(x)tan⁡3(x)+3(tan⁡2(x)+1)2e2log⁡(2)tan⁡e(x)tan⁡3(x)−3e(tan⁡2(x)+1)2tan⁡3(x)+2(tan⁡2(x)+1)2tan⁡3(x)+(tan⁡2(x)+1)2e2tan⁡3(x)+6e(tan⁡2(x)+1)log⁡(2)tan⁡e(x)tan⁡(x)−4(tan⁡2(x)+1)tan⁡(x)+6e(tan⁡2(x)+1)tan⁡(x)+4tan⁡(x))log⁡(2)tan⁡e(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
Derivada de y=2^tgx^e
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