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y=2^cos(tgx)
Derivada de la función/ (tgx)/ y=2^cos(tgx)

Derivada de y=2^cos(tgx)

Función f() - derivada -er orden en el punto
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Gráfico:

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Solución

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

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

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

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

      dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

    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:

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

    Como resultado de la secuencia de reglas:

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

  4. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-500500
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
  cos(tan(x)) /       2   \                   
-2           *\1 + tan (x)/*log(2)*sin(tan(x))
−2cos⁡(tan⁡(x))(tan⁡2(x)+1)log⁡(2)sin⁡(tan⁡(x))- 2^{\cos{\left(\tan{\left(x \right)} \right)}} \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(2 \right)} \sin{\left(\tan{\left(x \right)} \right)}
Simplificar
Segunda derivada [src] 
 cos(tan(x)) /       2   \ /  /       2   \                                         2         /       2   \       \       
2           *\1 + tan (x)/*\- \1 + tan (x)/*cos(tan(x)) - 2*sin(tan(x))*tan(x) + sin (tan(x))*\1 + tan (x)/*log(2)/*log(2)
2cos⁡(tan⁡(x))(tan⁡2(x)+1)((tan⁡2(x)+1)log⁡(2)sin⁡2(tan⁡(x))−(tan⁡2(x)+1)cos⁡(tan⁡(x))−2sin⁡(tan⁡(x))tan⁡(x))log⁡(2)2^{\cos{\left(\tan{\left(x \right)} \right)}} \left(\tan^{2}{\left(x \right)} + 1\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(2 \right)} \sin^{2}{\left(\tan{\left(x \right)} \right)} - \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(\tan{\left(x \right)} \right)} - 2 \sin{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}\right) \log{\left(2 \right)}
Simplificar
Tercera derivada [src] 
                           /             2                                                                                  2                                                                            2                                                                            \       
 cos(tan(x)) /       2   \ |/       2   \                     2                    /       2   \               /       2   \     2       3             /       2   \                        /       2   \                                        2         /       2   \              |       
2           *\1 + tan (x)/*\\1 + tan (x)/ *sin(tan(x)) - 4*tan (x)*sin(tan(x)) - 2*\1 + tan (x)/*sin(tan(x)) - \1 + tan (x)/ *log (2)*sin (tan(x)) - 6*\1 + tan (x)/*cos(tan(x))*tan(x) + 3*\1 + tan (x)/ *cos(tan(x))*log(2)*sin(tan(x)) + 6*sin (tan(x))*\1 + tan (x)/*log(2)*tan(x)/*log(2)
2cos⁡(tan⁡(x))(tan⁡2(x)+1)(−(tan⁡2(x)+1)2log⁡(2)2sin⁡3(tan⁡(x))+3(tan⁡2(x)+1)2log⁡(2)sin⁡(tan⁡(x))cos⁡(tan⁡(x))+(tan⁡2(x)+1)2sin⁡(tan⁡(x))+6(tan⁡2(x)+1)log⁡(2)sin⁡2(tan⁡(x))tan⁡(x)−2(tan⁡2(x)+1)sin⁡(tan⁡(x))−6(tan⁡2(x)+1)cos⁡(tan⁡(x))tan⁡(x)−4sin⁡(tan⁡(x))tan⁡2(x))log⁡(2)2^{\cos{\left(\tan{\left(x \right)} \right)}} \left(\tan^{2}{\left(x \right)} + 1\right) \left(- \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(2 \right)}^{2} \sin^{3}{\left(\tan{\left(x \right)} \right)} + 3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(2 \right)} \sin{\left(\tan{\left(x \right)} \right)} \cos{\left(\tan{\left(x \right)} \right)} + \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\tan{\left(x \right)} \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(2 \right)} \sin^{2}{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(\tan{\left(x \right)} \right)} - 6 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)} - 4 \sin{\left(\tan{\left(x \right)} \right)} \tan^{2}{\left(x \right)}\right) \log{\left(2 \right)}
Simplificar
Gráfico
Derivada de y=2^cos(tgx)
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