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Derivada de la función/ y=2^tgx/ y=2^tgx^3

Derivada de y=2^tgx^3

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

Ha introducido [src] 
    3   
 tan (x)
2
2tan⁡3(x)2^{\tan^{3}{\left(x \right)}} 
2^(tan(x)^3)
Solución detallada

  1. Sustituimos u=tan⁡3(x)u = \tan^{3}{\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⁡3(x)\frac{d}{d x} \tan^{3}{\left(x \right)}:

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

    2. Según el principio, aplicamos: u3u^{3} tenemos 3u23 u^{2}

    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:

      3(sin⁡2(x)+cos⁡2(x))tan⁡2(x)cos⁡2(x)\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

    Como resultado de la secuencia de reglas:

    3⋅2tan⁡3(x)(sin⁡2(x)+cos⁡2(x))log⁡(2)tan⁡2(x)cos⁡2(x)\frac{3 \cdot 2^{\tan^{3}{\left(x \right)}} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

  4. Simplificamos:

    3⋅2tan⁡3(x)log⁡(2)tan⁡2(x)cos⁡2(x)\frac{3 \cdot 2^{\tan^{3}{\left(x \right)}} \log{\left(2 \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

Respuesta:

3⋅2tan⁡3(x)log⁡(2)tan⁡2(x)cos⁡2(x)\frac{3 \cdot 2^{\tan^{3}{\left(x \right)}} \log{\left(2 \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

Primera derivada [src] 
    3                                  
 tan (x)    2    /         2   \       
2       *tan (x)*\3 + 3*tan (x)/*log(2)
2tan⁡3(x)(3tan⁡2(x)+3)log⁡(2)tan⁡2(x)2^{\tan^{3}{\left(x \right)}} \left(3 \tan^{2}{\left(x \right)} + 3\right) \log{\left(2 \right)} \tan^{2}{\left(x \right)}
Simplificar
Segunda derivada [src] 
      3                                                                                
   tan (x) /       2   \ /         2           3    /       2   \       \              
3*2       *\1 + tan (x)/*\2 + 4*tan (x) + 3*tan (x)*\1 + tan (x)/*log(2)/*log(2)*tan(x)
3⋅2tan⁡3(x)(tan⁡2(x)+1)(3(tan⁡2(x)+1)log⁡(2)tan⁡3(x)+4tan⁡2(x)+2)log⁡(2)tan⁡(x)3 \cdot 2^{\tan^{3}{\left(x \right)}} \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(2 \right)} \tan^{3}{\left(x \right)} + 4 \tan^{2}{\left(x \right)} + 2\right) \log{\left(2 \right)} \tan{\left(x \right)}
Simplificar
Tercera derivada [src] 
      3                  /               2                                                         2                                   2                                                 \       
   tan (x) /       2   \ |  /       2   \         4            2    /       2   \     /       2   \     2       6         /       2   \     3                   5    /       2   \       |       
3*2       *\1 + tan (x)/*\2*\1 + tan (x)/  + 4*tan (x) + 14*tan (x)*\1 + tan (x)/ + 9*\1 + tan (x)/ *log (2)*tan (x) + 18*\1 + tan (x)/ *tan (x)*log(2) + 18*tan (x)*\1 + tan (x)/*log(2)/*log(2)
3⋅2tan⁡3(x)(tan⁡2(x)+1)(9(tan⁡2(x)+1)2log⁡(2)2tan⁡6(x)+18(tan⁡2(x)+1)2log⁡(2)tan⁡3(x)+2(tan⁡2(x)+1)2+18(tan⁡2(x)+1)log⁡(2)tan⁡5(x)+14(tan⁡2(x)+1)tan⁡2(x)+4tan⁡4(x))log⁡(2)3 \cdot 2^{\tan^{3}{\left(x \right)}} \left(\tan^{2}{\left(x \right)} + 1\right) \left(9 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(2 \right)}^{2} \tan^{6}{\left(x \right)} + 18 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(2 \right)} \tan^{3}{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 18 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(2 \right)} \tan^{5}{\left(x \right)} + 14 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}\right) \log{\left(2 \right)}
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