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y=(cosx/x^2)+(x^2/sinx)

Derivada de y=(cosx/x^2)+(x^2/sinx)

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

Ha introducido [src]
            2  
cos(x)     x   
------ + ------
   2     sin(x)
  x            
x2sin(x)+cos(x)x2\frac{x^{2}}{\sin{\left(x \right)}} + \frac{\cos{\left(x \right)}}{x^{2}}
cos(x)/x^2 + x^2/sin(x)
Solución detallada
  1. diferenciamos x2sin(x)+cos(x)x2\frac{x^{2}}{\sin{\left(x \right)}} + \frac{\cos{\left(x \right)}}{x^{2}} miembro por miembro:

    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)=cos(x)f{\left(x \right)} = \cos{\left(x \right)} y g(x)=x2g{\left(x \right)} = x^{2}.

      Para calcular ddxf(x)\frac{d}{d x} f{\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)}

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

      1. Según el principio, aplicamos: x2x^{2} tenemos 2x2 x

      Ahora aplicamos la regla de la derivada de una divesión:

      x2sin(x)2xcos(x)x4\frac{- x^{2} \sin{\left(x \right)} - 2 x \cos{\left(x \right)}}{x^{4}}

    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)=x2f{\left(x \right)} = x^{2} y g(x)=sin(x)g{\left(x \right)} = \sin{\left(x \right)}.

      Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Según el principio, aplicamos: x2x^{2} tenemos 2x2 x

      Para calcular 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)}

      Ahora aplicamos la regla de la derivada de una divesión:

      x2cos(x)+2xsin(x)sin2(x)\frac{- x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}}{\sin^{2}{\left(x \right)}}

    Como resultado de: x2cos(x)+2xsin(x)sin2(x)+x2sin(x)2xcos(x)x4\frac{- x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{- x^{2} \sin{\left(x \right)} - 2 x \cos{\left(x \right)}}{x^{4}}

  2. Simplificamos:

    x5cos(x)sin2(x)+2x4sin(x)xsin(x)2cos(x)x3\frac{- \frac{x^{5} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 x^{4}}{\sin{\left(x \right)}} - x \sin{\left(x \right)} - 2 \cos{\left(x \right)}}{x^{3}}


Respuesta:

x5cos(x)sin2(x)+2x4sin(x)xsin(x)2cos(x)x3\frac{- \frac{x^{5} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 x^{4}}{\sin{\left(x \right)}} - x \sin{\left(x \right)} - 2 \cos{\left(x \right)}}{x^{3}}

Gráfica
02468-8-6-4-2-1010-100000100000
Primera derivada [src]
                                2       
  sin(x)   2*cos(x)    2*x     x *cos(x)
- ------ - -------- + ------ - ---------
     2         3      sin(x)       2    
    x         x                 sin (x) 
x2cos(x)sin2(x)+2xsin(x)sin(x)x22cos(x)x3- \frac{x^{2} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 x}{\sin{\left(x \right)}} - \frac{\sin{\left(x \right)}}{x^{2}} - \frac{2 \cos{\left(x \right)}}{x^{3}}
Segunda derivada [src]
            2                                                    2    2   
  2        x      cos(x)   4*sin(x)   6*cos(x)   4*x*cos(x)   2*x *cos (x)
------ + ------ - ------ + -------- + -------- - ---------- + ------------
sin(x)   sin(x)      2         3          4          2             3      
                    x         x          x        sin (x)       sin (x)   
x2sin(x)+2x2cos2(x)sin3(x)4xcos(x)sin2(x)+2sin(x)cos(x)x2+4sin(x)x3+6cos(x)x4\frac{x^{2}}{\sin{\left(x \right)}} + \frac{2 x^{2} \cos^{2}{\left(x \right)}}{\sin^{3}{\left(x \right)}} - \frac{4 x \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2}{\sin{\left(x \right)}} - \frac{\cos{\left(x \right)}}{x^{2}} + \frac{4 \sin{\left(x \right)}}{x^{3}} + \frac{6 \cos{\left(x \right)}}{x^{4}}
Tercera derivada [src]
                                                                   2    3         2                  2   
sin(x)   24*cos(x)   18*sin(x)   6*cos(x)    6*x     6*cos(x)   6*x *cos (x)   5*x *cos(x)   12*x*cos (x)
------ - --------- - --------- - -------- + ------ + -------- - ------------ - ----------- + ------------
   2          5           4         2       sin(x)       3           4              2             3      
  x          x           x       sin (x)                x         sin (x)        sin (x)       sin (x)   
5x2cos(x)sin2(x)6x2cos3(x)sin4(x)+6xsin(x)+12xcos2(x)sin3(x)6cos(x)sin2(x)+sin(x)x2+6cos(x)x318sin(x)x424cos(x)x5- \frac{5 x^{2} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{6 x^{2} \cos^{3}{\left(x \right)}}{\sin^{4}{\left(x \right)}} + \frac{6 x}{\sin{\left(x \right)}} + \frac{12 x \cos^{2}{\left(x \right)}}{\sin^{3}{\left(x \right)}} - \frac{6 \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{x^{2}} + \frac{6 \cos{\left(x \right)}}{x^{3}} - \frac{18 \sin{\left(x \right)}}{x^{4}} - \frac{24 \cos{\left(x \right)}}{x^{5}}
Gráfico
Derivada de y=(cosx/x^2)+(x^2/sinx)