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y=(x(tgx))/(1+x^2)
Derivada de la función/ (tgx)/ 1+x^2/ y=(x(tgx))/(1+x^2)

Derivada de y=(x(tgx))/(1+x^2)

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

Ha introducido [src] 
x*tan(x)
--------
      2 
 1 + x
xtan⁡(x)x2+1\frac{x \tan{\left(x \right)}}{x^{2} + 1} 
(x*tan(x))/(1 + 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)=xtan⁡(x)f{\left(x \right)} = x \tan{\left(x \right)} y g(x)=x2+1g{\left(x \right)} = x^{2} + 1.

    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)=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))cos⁡2(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)}

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

    1. diferenciamos x2+1x^{2} + 1 miembro por miembro:

      1. La derivada de una constante 11 es igual a cero.

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

      Como resultado de: 2x2 x

    Ahora aplicamos la regla de la derivada de una divesión:

    −2x2tan⁡(x)+(x2+1)(x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)+tan⁡(x))(x2+1)2\frac{- 2 x^{2} \tan{\left(x \right)} + \left(x^{2} + 1\right) \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)}{\left(x^{2} + 1\right)^{2}}

  2. Simplificamos:

    x3−x2sin⁡(2x)2+x+sin⁡(2x)2(x2+1)2cos⁡2(x)\frac{x^{3} - \frac{x^{2} \sin{\left(2 x \right)}}{2} + x + \frac{\sin{\left(2 x \right)}}{2}}{\left(x^{2} + 1\right)^{2} \cos^{2}{\left(x \right)}}

Respuesta:

x3−x2sin⁡(2x)2+x+sin⁡(2x)2(x2+1)2cos⁡2(x)\frac{x^{3} - \frac{x^{2} \sin{\left(2 x \right)}}{2} + x + \frac{\sin{\left(2 x \right)}}{2}}{\left(x^{2} + 1\right)^{2} \cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-500500
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Trazar el gráfico f'(x)
Primera derivada [src] 
  /       2   \               2       
x*\1 + tan (x)/ + tan(x)   2*x *tan(x)
------------------------ - -----------
              2                     2 
         1 + x              /     2\  
                            \1 + x /
−2x2tan⁡(x)(x2+1)2+x(tan⁡2(x)+1)+tan⁡(x)x2+1- \frac{2 x^{2} \tan{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}}{x^{2} + 1}
Simplificar
Segunda derivada [src] 
  /                                                                          /         2 \       \
  |                                                                          |      4*x  |       |
  |                                                                        x*|-1 + ------|*tan(x)|
  |                                           /  /       2   \         \     |          2|       |
  |       2        /       2   \          2*x*\x*\1 + tan (x)/ + tan(x)/     \     1 + x /       |
2*|1 + tan (x) + x*\1 + tan (x)/*tan(x) - ------------------------------ + ----------------------|
  |                                                        2                            2        |
  \                                                   1 + x                        1 + x         /
--------------------------------------------------------------------------------------------------
                                                   2                                              
                                              1 + x
2(x(tan⁡2(x)+1)tan⁡(x)−2x(x(tan⁡2(x)+1)+tan⁡(x))x2+1+x(4x2x2+1−1)tan⁡(x)x2+1+tan⁡2(x)+1)x2+1\frac{2 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2 x \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right)}{x^{2} + 1} + \frac{x \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right) \tan{\left(x \right)}}{x^{2} + 1} + \tan^{2}{\left(x \right)} + 1\right)}{x^{2} + 1}
Simplificar
Tercera derivada [src] 
  /                                                                                              /         2 \                                    /         2 \       \
  |                                                                                              |      4*x  | /  /       2   \         \       2 |      2*x  |       |
  |                                                                                            3*|-1 + ------|*\x*\1 + tan (x)/ + tan(x)/   12*x *|-1 + ------|*tan(x)|
  |                                                   /       2        /       2   \       \     |          2|                                    |          2|       |
  |/       2   \ /             /         2   \\   6*x*\1 + tan (x) + x*\1 + tan (x)/*tan(x)/     \     1 + x /                                    \     1 + x /       |
2*|\1 + tan (x)/*\3*tan(x) + x*\1 + 3*tan (x)// - ------------------------------------------ + ------------------------------------------ - --------------------------|
  |                                                                      2                                            2                                     2         |
  |                                                                 1 + x                                        1 + x                              /     2\          |
  \                                                                                                                                                 \1 + x /          /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                      2                                                                                
                                                                                 1 + x
2(−12x2(2x2x2+1−1)tan⁡(x)(x2+1)2−6x(x(tan⁡2(x)+1)tan⁡(x)+tan⁡2(x)+1)x2+1+(x(3tan⁡2(x)+1)+3tan⁡(x))(tan⁡2(x)+1)+3(x(tan⁡2(x)+1)+tan⁡(x))(4x2x2+1−1)x2+1)x2+1\frac{2 \left(- \frac{12 x^{2} \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right) \tan{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} - \frac{6 x \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)}{x^{2} + 1} + \left(x \left(3 \tan^{2}{\left(x \right)} + 1\right) + 3 \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \frac{3 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1}\right)}{x^{2} + 1}
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Gráfico
Derivada de y=(x(tgx))/(1+x^2)
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