Halla la derivada y' = f'(x) = y=2cos²xtanx (y es igual a 2 coseno de al cuadrado x tangente de x) - funciones. Hallemos el valor de la derivada de la función en el punto. [¡Hay una RESPUESTA!] online
Sr Examen

Derivada de y=2cos^2xtanx

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

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Solución

Ha introducido [src]
     2          
2*cos (x*tan(x))
2cos2(xtan(x))2 \cos^{2}{\left(x \tan{\left(x \right)} \right)}
2*cos(x*tan(x))^2
Solución detallada
  1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.

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

    2. Según el principio, aplicamos: u2u^{2} tenemos 2u2 u

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

      1. Sustituimos u=xtan(x)u = x \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 ddxxtan(x)\frac{d}{d x} x \tan{\left(x \right)}:

        1. Se aplica la regla de la derivada de una multiplicación:

          ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=xf{\left(x \right)} = x; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

          1. Según el principio, aplicamos: xx tenemos 11

          g(x)=tan(x)g{\left(x \right)} = \tan{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\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:

            sin2(x)+cos2(x)cos2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

          Como resultado de: x(sin2(x)+cos2(x))cos2(x)+tan(x)\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}

        Como resultado de la secuencia de reglas:

        (x(sin2(x)+cos2(x))cos2(x)+tan(x))sin(xtan(x))- \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}\right) \sin{\left(x \tan{\left(x \right)} \right)}

      Como resultado de la secuencia de reglas:

      2(x(sin2(x)+cos2(x))cos2(x)+tan(x))sin(xtan(x))cos(xtan(x))- 2 \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}\right) \sin{\left(x \tan{\left(x \right)} \right)} \cos{\left(x \tan{\left(x \right)} \right)}

    Entonces, como resultado: 4(x(sin2(x)+cos2(x))cos2(x)+tan(x))sin(xtan(x))cos(xtan(x))- 4 \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}\right) \sin{\left(x \tan{\left(x \right)} \right)} \cos{\left(x \tan{\left(x \right)} \right)}

  2. Simplificamos:

    4(xcos2(x)+tan(x))sin(xtan(x))cos(xtan(x))- 4 \left(\frac{x}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}\right) \sin{\left(x \tan{\left(x \right)} \right)} \cos{\left(x \tan{\left(x \right)} \right)}


Respuesta:

4(xcos2(x)+tan(x))sin(xtan(x))cos(xtan(x))- 4 \left(\frac{x}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}\right) \sin{\left(x \tan{\left(x \right)} \right)} \cos{\left(x \tan{\left(x \right)} \right)}

Gráfica
02468-8-6-4-2-1010-50005000
Primera derivada [src]
   /  /       2   \         \                            
-4*\x*\1 + tan (x)/ + tan(x)/*cos(x*tan(x))*sin(x*tan(x))
4(x(tan2(x)+1)+tan(x))sin(xtan(x))cos(xtan(x))- 4 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \sin{\left(x \tan{\left(x \right)} \right)} \cos{\left(x \tan{\left(x \right)} \right)}
Segunda derivada [src]
  /                          2                                            2                                                                                      \
  |/  /       2   \         \     2             /  /       2   \         \     2               /       2        /       2   \       \                            |
4*\\x*\1 + tan (x)/ + tan(x)/ *sin (x*tan(x)) - \x*\1 + tan (x)/ + tan(x)/ *cos (x*tan(x)) - 2*\1 + tan (x) + x*\1 + tan (x)/*tan(x)/*cos(x*tan(x))*sin(x*tan(x))/
4((x(tan2(x)+1)+tan(x))2sin2(xtan(x))(x(tan2(x)+1)+tan(x))2cos2(xtan(x))2(x(tan2(x)+1)tan(x)+tan2(x)+1)sin(xtan(x))cos(xtan(x)))4 \left(\left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right)^{2} \sin^{2}{\left(x \tan{\left(x \right)} \right)} - \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right)^{2} \cos^{2}{\left(x \tan{\left(x \right)} \right)} - 2 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \tan{\left(x \right)} \right)} \cos{\left(x \tan{\left(x \right)} \right)}\right)
Tercera derivada [src]
  /                                                                                                                   3                                                                                                                                                                                                        \
  |       2           /  /       2   \         \ /       2        /       2   \       \     /  /       2   \         \                                     2           /  /       2   \         \ /       2        /       2   \       \   /       2   \ /             /       2   \          2   \                            |
8*\- 3*cos (x*tan(x))*\x*\1 + tan (x)/ + tan(x)/*\1 + tan (x) + x*\1 + tan (x)/*tan(x)/ + 2*\x*\1 + tan (x)/ + tan(x)/ *cos(x*tan(x))*sin(x*tan(x)) + 3*sin (x*tan(x))*\x*\1 + tan (x)/ + tan(x)/*\1 + tan (x) + x*\1 + tan (x)/*tan(x)/ - \1 + tan (x)/*\3*tan(x) + x*\1 + tan (x)/ + 2*x*tan (x)/*cos(x*tan(x))*sin(x*tan(x))/
8(2(x(tan2(x)+1)+tan(x))3sin(xtan(x))cos(xtan(x))+3(x(tan2(x)+1)+tan(x))(x(tan2(x)+1)tan(x)+tan2(x)+1)sin2(xtan(x))3(x(tan2(x)+1)+tan(x))(x(tan2(x)+1)tan(x)+tan2(x)+1)cos2(xtan(x))(tan2(x)+1)(x(tan2(x)+1)+2xtan2(x)+3tan(x))sin(xtan(x))cos(xtan(x)))8 \left(2 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right)^{3} \sin{\left(x \tan{\left(x \right)} \right)} \cos{\left(x \tan{\left(x \right)} \right)} + 3 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(x \tan{\left(x \right)} \right)} - 3 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right) \cos^{2}{\left(x \tan{\left(x \right)} \right)} - \left(\tan^{2}{\left(x \right)} + 1\right) \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 x \tan^{2}{\left(x \right)} + 3 \tan{\left(x \right)}\right) \sin{\left(x \tan{\left(x \right)} \right)} \cos{\left(x \tan{\left(x \right)} \right)}\right)
Gráfico
Derivada de y=2cos^2xtanx