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y=sinx/(cos^2)x

Derivada de y=sinx/(cos^2)x

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

Ha introducido [src]
 sin(x)  
-------*x
   2     
cos (x)  
xsin(x)cos2(x)x \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}
(sin(x)/cos(x)^2)*x
Solución detallada
  1. 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)=xsin(x)f{\left(x \right)} = x \sin{\left(x \right)} y g(x)=cos2(x)g{\left(x \right)} = \cos^{2}{\left(x \right)}.

    Para calcular 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)=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)=sin(x)g{\left(x \right)} = \sin{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\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)}

      Como resultado de: xcos(x)+sin(x)x \cos{\left(x \right)} + \sin{\left(x \right)}

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

    Ahora aplicamos la regla de la derivada de una divesión:

    2xsin2(x)cos(x)+(xcos(x)+sin(x))cos2(x)cos4(x)\frac{2 x \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)}}

  2. Simplificamos:

    xcos(2x)+3x+sin(2x)2cos3(x)\frac{- x \cos{\left(2 x \right)} + 3 x + \sin{\left(2 x \right)}}{2 \cos^{3}{\left(x \right)}}


Respuesta:

xcos(2x)+3x+sin(2x)2cos3(x)\frac{- x \cos{\left(2 x \right)} + 3 x + \sin{\left(2 x \right)}}{2 \cos^{3}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-500000500000
Primera derivada [src]
  /               2   \          
  | cos(x)   2*sin (x)|    sin(x)
x*|------- + ---------| + -------
  |   2          3    |      2   
  \cos (x)    cos (x) /   cos (x)
x(2sin2(x)cos3(x)+cos(x)cos2(x))+sin(x)cos2(x)x \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{\cos{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) + \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}
Segunda derivada [src]
                  /         2   \       
                  |    6*sin (x)|       
                x*|5 + ---------|*sin(x)
         2        |        2    |       
    4*sin (x)     \     cos (x) /       
2 + --------- + ------------------------
        2                cos(x)         
     cos (x)                            
----------------------------------------
                 cos(x)                 
x(6sin2(x)cos2(x)+5)sin(x)cos(x)+4sin2(x)cos2(x)+2cos(x)\frac{\frac{x \left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{4 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2}{\cos{\left(x \right)}}
Tercera derivada [src]
  /                           /         2   \\     /         2   \       
  |                      2    |    3*sin (x)||     |    6*sin (x)|       
  |                 8*sin (x)*|2 + ---------||   3*|5 + ---------|*sin(x)
  |          2                |        2    ||     |        2    |       
  |    12*sin (x)             \     cos (x) /|     \     cos (x) /       
x*|5 + ---------- + -------------------------| + ------------------------
  |        2                    2            |            cos(x)         
  \     cos (x)              cos (x)         /                           
-------------------------------------------------------------------------
                                  cos(x)                                 
x(8(3sin2(x)cos2(x)+2)sin2(x)cos2(x)+12sin2(x)cos2(x)+5)+3(6sin2(x)cos2(x)+5)sin(x)cos(x)cos(x)\frac{x \left(\frac{8 \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{12 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) + \frac{3 \left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin{\left(x \right)}}{\cos{\left(x \right)}}}{\cos{\left(x \right)}}
Gráfico
Derivada de y=sinx/(cos^2)x