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y=e^cos^2x*xtgx
Derivada de la función/ cos^2/ y=e^c/ y=e^cos^2x*xtgx

Derivada de y=e^cos^2x*xtgx

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

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

  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)=ecos⁡2(x)xf{\left(x \right)} = e^{\cos^{2}{\left(x \right)}} x; calculamos ddxf(x)\frac{d}{d x} f{\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)=ecos⁡2(x)f{\left(x \right)} = e^{\cos^{2}{\left(x \right)}}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Sustituimos u=cos⁡2(x)u = \cos^{2}{\left(x \right)}.

      2. Derivado eue^{u} es.

      3. Luego se aplica una cadena de reglas. Multiplicamos por ddxcos⁡2(x)\frac{d}{d x} \cos^{2}{\left(x \right)}:

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

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

        3. Luego se aplica una cadena de reglas. Multiplicamos por ddxcos⁡(x)\frac{d}{d x} \cos{\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)}

          Como resultado de la secuencia de reglas:

          −2sin⁡(x)cos⁡(x)- 2 \sin{\left(x \right)} \cos{\left(x \right)}

        Como resultado de la secuencia de reglas:

        −2ecos⁡2(x)sin⁡(x)cos⁡(x)- 2 e^{\cos^{2}{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}

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

      1. Según el principio, aplicamos: xx tenemos 11

      Como resultado de: ecos⁡2(x)−2xecos⁡2(x)sin⁡(x)cos⁡(x)e^{\cos^{2}{\left(x \right)}} - 2 x e^{\cos^{2}{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}

    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:

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

  2. Simplificamos:

    (x−(xsin⁡(2x)−1)cos⁡2(x)tan⁡(x))ecos⁡2(x)cos⁡2(x)\frac{\left(x - \left(x \sin{\left(2 x \right)} - 1\right) \cos^{2}{\left(x \right)} \tan{\left(x \right)}\right) e^{\cos^{2}{\left(x \right)}}}{\cos^{2}{\left(x \right)}}

Respuesta:

(x−(xsin⁡(2x)−1)cos⁡2(x)tan⁡(x))ecos⁡2(x)cos⁡2(x)\frac{\left(x - \left(x \sin{\left(2 x \right)} - 1\right) \cos^{2}{\left(x \right)} \tan{\left(x \right)}\right) e^{\cos^{2}{\left(x \right)}}}{\cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-1000010000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
/    2                     2          \                              2   
| cos (x)               cos (x)       |            /       2   \  cos (x)
\E        - 2*x*cos(x)*e       *sin(x)/*tan(x) + x*\1 + tan (x)/*e
x(tan⁡2(x)+1)ecos⁡2(x)+(ecos⁡2(x)−2xecos⁡2(x)sin⁡(x)cos⁡(x))tan⁡(x)x \left(\tan^{2}{\left(x \right)} + 1\right) e^{\cos^{2}{\left(x \right)}} + \left(e^{\cos^{2}{\left(x \right)}} - 2 x e^{\cos^{2}{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}\right) \tan{\left(x \right)}
Simplificar
Segunda derivada [src] 
                                                                                                                                               2   
  //  /   2         2           2       2   \                  \          /       2   \                              /       2   \       \  cos (x)
2*\\x*\sin (x) - cos (x) + 2*cos (x)*sin (x)/ - 2*cos(x)*sin(x)/*tan(x) - \1 + tan (x)/*(-1 + 2*x*cos(x)*sin(x)) + x*\1 + tan (x)/*tan(x)/*e
2(x(tan⁡2(x)+1)tan⁡(x)+(x(2sin⁡2(x)cos⁡2(x)+sin⁡2(x)−cos⁡2(x))−2sin⁡(x)cos⁡(x))tan⁡(x)−(2xsin⁡(x)cos⁡(x)−1)(tan⁡2(x)+1))ecos⁡2(x)2 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(x \left(2 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \tan{\left(x \right)} - \left(2 x \sin{\left(x \right)} \cos{\left(x \right)} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)\right) e^{\cos^{2}{\left(x \right)}}
Simplificar
Tercera derivada [src] 
                                                                                                                                                                                                                                                                                                        2   
  /  /       2           2           2       2          /          2           2           2       2   \              \            /       2   \ /  /   2         2           2       2   \                  \     /       2   \ /         2   \     /       2   \                                \  cos (x)
2*\- \- 3*sin (x) + 3*cos (x) - 6*cos (x)*sin (x) + 2*x*\-2 - 3*cos (x) + 3*sin (x) + 2*cos (x)*sin (x)/*cos(x)*sin(x)/*tan(x) + 3*\1 + tan (x)/*\x*\sin (x) - cos (x) + 2*cos (x)*sin (x)/ - 2*cos(x)*sin(x)/ + x*\1 + tan (x)/*\1 + 3*tan (x)/ - 3*\1 + tan (x)/*(-1 + 2*x*cos(x)*sin(x))*tan(x)/*e
2(x(tan⁡2(x)+1)(3tan⁡2(x)+1)+3(x(2sin⁡2(x)cos⁡2(x)+sin⁡2(x)−cos⁡2(x))−2sin⁡(x)cos⁡(x))(tan⁡2(x)+1)−3(2xsin⁡(x)cos⁡(x)−1)(tan⁡2(x)+1)tan⁡(x)−(2x(2sin⁡2(x)cos⁡2(x)+3sin⁡2(x)−3cos⁡2(x)−2)sin⁡(x)cos⁡(x)−6sin⁡2(x)cos⁡2(x)−3sin⁡2(x)+3cos⁡2(x))tan⁡(x))ecos⁡2(x)2 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + 3 \left(x \left(2 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) - 3 \left(2 x \sin{\left(x \right)} \cos{\left(x \right)} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \left(2 x \left(2 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + 3 \sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)} - 2\right) \sin{\left(x \right)} \cos{\left(x \right)} - 6 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} - 3 \sin^{2}{\left(x \right)} + 3 \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}\right) e^{\cos^{2}{\left(x \right)}}
Simplificar
Gráfico
Derivada de y=e^cos^2x*xtgx
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