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y=(1+cot^2x)/cotx

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

Función f() - derivada -er orden en el punto
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Gráfico:

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Solución

Ha introducido [src]
       2   
1 + cot (x)
-----------
   cot(x)  
cot2(x)+1cot(x)\frac{\cot^{2}{\left(x \right)} + 1}{\cot{\left(x \right)}}
(1 + cot(x)^2)/cot(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)=cot2(x)+1f{\left(x \right)} = \cot^{2}{\left(x \right)} + 1 y g(x)=cot(x)g{\left(x \right)} = \cot{\left(x \right)}.

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

    1. diferenciamos cot2(x)+1\cot^{2}{\left(x \right)} + 1 miembro por miembro:

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

      2. Sustituimos u=cot(x)u = \cot{\left(x \right)}.

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

      4. Luego se aplica una cadena de reglas. Multiplicamos por ddxcot(x)\frac{d}{d x} \cot{\left(x \right)}:

        1. Hay varias formas de calcular esta derivada.

          Method #1

          1. Reescribimos las funciones para diferenciar:

            cot(x)=1tan(x)\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}

          2. Sustituimos u=tan(x)u = \tan{\left(x \right)}.

          3. Según el principio, aplicamos: 1u\frac{1}{u} tenemos 1u2- \frac{1}{u^{2}}

          4. Luego se aplica una cadena de reglas. Multiplicamos por ddxtan(x)\frac{d}{d x} \tan{\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)}}

            Como resultado de la secuencia de reglas:

            sin2(x)+cos2(x)cos2(x)tan2(x)- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}

          Method #2

          1. Reescribimos las funciones para diferenciar:

            cot(x)=cos(x)sin(x)\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\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)=cos(x)f{\left(x \right)} = \cos{\left(x \right)} 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. 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. 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:

            sin2(x)cos2(x)sin2(x)\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}

        Como resultado de la secuencia de reglas:

        2(sin2(x)+cos2(x))cot(x)cos2(x)tan2(x)- \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \cot{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}

      Como resultado de: 2(sin2(x)+cos2(x))cot(x)cos2(x)tan2(x)- \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \cot{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}

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

    1. ddxcot(x)=1sin2(x)\frac{d}{d x} \cot{\left(x \right)} = - \frac{1}{\sin^{2}{\left(x \right)}}

    Ahora aplicamos la regla de la derivada de una divesión:

    (sin2(x)+cos2(x))(cot2(x)+1)cos2(x)tan2(x)2(sin2(x)+cos2(x))cot2(x)cos2(x)tan2(x)cot2(x)\frac{\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}}{\cot^{2}{\left(x \right)}}

  2. Simplificamos:

    11tan2(x)cos2(x)\frac{1 - \frac{1}{\tan^{2}{\left(x \right)}}}{\cos^{2}{\left(x \right)}}


Respuesta:

11tan2(x)cos2(x)\frac{1 - \frac{1}{\tan^{2}{\left(x \right)}}}{\cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-1000010000
Primera derivada [src]
                              2
                 /       2   \ 
          2      \1 + cot (x)/ 
-2 - 2*cot (x) + --------------
                       2       
                    cot (x)    
(cot2(x)+1)2cot2(x)2cot2(x)2\frac{\left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{2}{\left(x \right)}} - 2 \cot^{2}{\left(x \right)} - 2
Segunda derivada [src]
                /                             /            2   \\
  /       2   \ |        2      /       2   \ |     1 + cot (x)||
2*\1 + cot (x)/*|-1 + cot (x) + \1 + cot (x)/*|-1 + -----------||
                |                             |          2     ||
                \                             \       cot (x)  //
-----------------------------------------------------------------
                              cot(x)                             
2(cot2(x)+1)((cot2(x)+1cot2(x)1)(cot2(x)+1)+cot2(x)1)cot(x)\frac{2 \left(\cot^{2}{\left(x \right)} + 1\right) \left(\left(\frac{\cot^{2}{\left(x \right)} + 1}{\cot^{2}{\left(x \right)}} - 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) + \cot^{2}{\left(x \right)} - 1\right)}{\cot{\left(x \right)}}
Tercera derivada [src]
                /                                                                      2                  3                                  \
                |                                  /            2   \     /       2   \      /       2   \      /       2   \ /         2   \|
  /       2   \ |           2        /       2   \ |     1 + cot (x)|   5*\1 + cot (x)/    3*\1 + cot (x)/    3*\1 + cot (x)/*\1 + 3*cot (x)/|
2*\1 + cot (x)/*|-6 - 10*cot (x) - 6*\1 + cot (x)/*|-1 + -----------| - ---------------- + ---------------- + -------------------------------|
                |                                  |          2     |          2                  4                          2               |
                \                                  \       cot (x)  /       cot (x)            cot (x)                    cot (x)            /
2(cot2(x)+1)(6(cot2(x)+1cot2(x)1)(cot2(x)+1)+3(cot2(x)+1)3cot4(x)5(cot2(x)+1)2cot2(x)+3(cot2(x)+1)(3cot2(x)+1)cot2(x)10cot2(x)6)2 \left(\cot^{2}{\left(x \right)} + 1\right) \left(- 6 \left(\frac{\cot^{2}{\left(x \right)} + 1}{\cot^{2}{\left(x \right)}} - 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) + \frac{3 \left(\cot^{2}{\left(x \right)} + 1\right)^{3}}{\cot^{4}{\left(x \right)}} - \frac{5 \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{2}{\left(x \right)}} + \frac{3 \left(\cot^{2}{\left(x \right)} + 1\right) \left(3 \cot^{2}{\left(x \right)} + 1\right)}{\cot^{2}{\left(x \right)}} - 10 \cot^{2}{\left(x \right)} - 6\right)
Gráfico
Derivada de y=(1+cot^2x)/cotx