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y=tgx/x^2

Derivada de y=tgx/x^2

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

Ha introducido [src]
tan(x)
------
   2  
  x   
tan(x)x2\frac{\tan{\left(x \right)}}{x^{2}}
tan(x)/x^2
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)=tan(x)f{\left(x \right)} = \tan{\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. 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)}}

    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:

    x2(sin2(x)+cos2(x))cos2(x)2xtan(x)x4\frac{\frac{x^{2} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - 2 x \tan{\left(x \right)}}{x^{4}}

  2. Simplificamos:

    xsin(2x)x3cos2(x)\frac{x - \sin{\left(2 x \right)}}{x^{3} \cos^{2}{\left(x \right)}}


Respuesta:

xsin(2x)x3cos2(x)\frac{x - \sin{\left(2 x \right)}}{x^{3} \cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-250250
Primera derivada [src]
       2              
1 + tan (x)   2*tan(x)
----------- - --------
      2           3   
     x           x    
tan2(x)+1x22tan(x)x3\frac{\tan^{2}{\left(x \right)} + 1}{x^{2}} - \frac{2 \tan{\left(x \right)}}{x^{3}}
Segunda derivada [src]
  /                         /       2   \           \
  |/       2   \          2*\1 + tan (x)/   3*tan(x)|
2*|\1 + tan (x)/*tan(x) - --------------- + --------|
  |                              x              2   |
  \                                            x    /
-----------------------------------------------------
                           2                         
                          x                          
2((tan2(x)+1)tan(x)2(tan2(x)+1)x+3tan(x)x2)x2\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{3 \tan{\left(x \right)}}{x^{2}}\right)}{x^{2}}
Tercera derivada [src]
  /                                              /       2   \     /       2   \       \
  |/       2   \ /         2   \   12*tan(x)   9*\1 + tan (x)/   6*\1 + tan (x)/*tan(x)|
2*|\1 + tan (x)/*\1 + 3*tan (x)/ - --------- + --------------- - ----------------------|
  |                                     3              2                   x           |
  \                                    x              x                                /
----------------------------------------------------------------------------------------
                                            2                                           
                                           x                                            
2((tan2(x)+1)(3tan2(x)+1)6(tan2(x)+1)tan(x)x+9(tan2(x)+1)x212tan(x)x3)x2\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} + \frac{9 \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2}} - \frac{12 \tan{\left(x \right)}}{x^{3}}\right)}{x^{2}}
Gráfico
Derivada de y=tgx/x^2