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y=ctgx+x/(1-xctgx)
Derivada de la función/ y=ctg/ xctgx/ y=ctgx+x/(1-xctgx)

Derivada de y=ctgx+x/(1-xctgx)

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

Ha introducido [src] 
              x      
cot(x) + ------------
         1 - x*cot(x)
x−xcot⁡(x)+1+cot⁡(x)\frac{x}{- x \cot{\left(x \right)} + 1} + \cot{\left(x \right)} 
cot(x) + x/(1 - x*cot(x))
Solución detallada

  1. diferenciamos x−xcot⁡(x)+1+cot⁡(x)\frac{x}{- x \cot{\left(x \right)} + 1} + \cot{\left(x \right)} miembro por miembro:

    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:

          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 la secuencia de reglas:

        −sin⁡2(x)+cos⁡2(x)cos⁡2(x)tan⁡2(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:

        −sin⁡2(x)−cos⁡2(x)sin⁡2(x)\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\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)=xf{\left(x \right)} = x y g(x)=−xcot⁡(x)+1g{\left(x \right)} = - x \cot{\left(x \right)} + 1.

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

      1. Según el principio, aplicamos: xx tenemos 11

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

      1. diferenciamos −xcot⁡(x)+1- x \cot{\left(x \right)} + 1 miembro por miembro:

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

        2. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.

          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)=cot⁡(x)g{\left(x \right)} = \cot{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

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

          Entonces, como resultado: x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)−cot⁡(x)\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} - \cot{\left(x \right)}

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

      Ahora aplicamos la regla de la derivada de una divesión:

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

    Como resultado de: −sin⁡2(x)+cos⁡2(x)cos⁡2(x)tan⁡2(x)+−x(x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)−cot⁡(x))−xcot⁡(x)+1(−xcot⁡(x)+1)2- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \frac{- x \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} - \cot{\left(x \right)}\right) - x \cot{\left(x \right)} + 1}{\left(- x \cot{\left(x \right)} + 1\right)^{2}}

  2. Simplificamos:

    −x2sin⁡2(x)cos⁡2(x)+2xsin⁡(x)cos⁡(x)−1(x−tan⁡(x))2\frac{- \frac{x^{2}}{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}} + \frac{2 x}{\sin{\left(x \right)} \cos{\left(x \right)}} - 1}{\left(x - \tan{\left(x \right)}\right)^{2}}

Respuesta:

−x2sin⁡2(x)cos⁡2(x)+2xsin⁡(x)cos⁡(x)−1(x−tan⁡(x))2\frac{- \frac{x^{2}}{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}} + \frac{2 x}{\sin{\left(x \right)} \cos{\left(x \right)}} - 1}{\left(x - \tan{\left(x \right)}\right)^{2}}

Gráfica
02468-8-6-4-2-1010-1000010000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
                                /  /        2   \         \
          1            2      x*\x*\-1 - cot (x)/ + cot(x)/
-1 + ------------ - cot (x) + -----------------------------
     1 - x*cot(x)                                  2       
                                     (1 - x*cot(x))
x(x(−cot⁡2(x)−1)+cot⁡(x))(−xcot⁡(x)+1)2−cot⁡2(x)−1+1−xcot⁡(x)+1\frac{x \left(x \left(- \cot^{2}{\left(x \right)} - 1\right) + \cot{\left(x \right)}\right)}{\left(- x \cot{\left(x \right)} + 1\right)^{2}} - \cot^{2}{\left(x \right)} - 1 + \frac{1}{- x \cot{\left(x \right)} + 1}
Simplificar
Segunda derivada [src] 
  /                                                                                2                                           \
  |                                   /       2   \     /            /       2   \\      /       2        /       2   \       \|
  |/       2   \          -cot(x) + x*\1 + cot (x)/   x*\-cot(x) + x*\1 + cot (x)//    x*\1 + cot (x) - x*\1 + cot (x)/*cot(x)/|
2*|\1 + cot (x)/*cot(x) - ------------------------- - ------------------------------ - ----------------------------------------|
  |                                           2                             3                                     2            |
  \                            (-1 + x*cot(x))               (-1 + x*cot(x))                       (-1 + x*cot(x))             /
2(−x(x(cot⁡2(x)+1)−cot⁡(x))2(xcot⁡(x)−1)3−x(−x(cot⁡2(x)+1)cot⁡(x)+cot⁡2(x)+1)(xcot⁡(x)−1)2−x(cot⁡2(x)+1)−cot⁡(x)(xcot⁡(x)−1)2+(cot⁡2(x)+1)cot⁡(x))2 \left(- \frac{x \left(x \left(\cot^{2}{\left(x \right)} + 1\right) - \cot{\left(x \right)}\right)^{2}}{\left(x \cot{\left(x \right)} - 1\right)^{3}} - \frac{x \left(- x \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 1\right)}{\left(x \cot{\left(x \right)} - 1\right)^{2}} - \frac{x \left(\cot^{2}{\left(x \right)} + 1\right) - \cot{\left(x \right)}}{\left(x \cot{\left(x \right)} - 1\right)^{2}} + \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right)
Simplificar
Tercera derivada [src] 
   /                                                                        2                                                                             3                                                                                                                                       \
   |             2                               /            /       2   \\      /       2        /       2   \       \       /            /       2   \\      /       2   \ /              /       2   \          2   \       /            /       2   \\ /       2        /       2   \       \|
   |/       2   \         2    /       2   \   3*\-cot(x) + x*\1 + cot (x)//    3*\1 + cot (x) - x*\1 + cot (x)/*cot(x)/   3*x*\-cot(x) + x*\1 + cot (x)//    x*\1 + cot (x)/*\-3*cot(x) + x*\1 + cot (x)/ + 2*x*cot (x)/   6*x*\-cot(x) + x*\1 + cot (x)//*\1 + cot (x) - x*\1 + cot (x)/*cot(x)/|
-2*|\1 + cot (x)/  + 2*cot (x)*\1 + cot (x)/ + ------------------------------ + ---------------------------------------- + -------------------------------- + ----------------------------------------------------------- + ----------------------------------------------------------------------|
   |                                                                 3                                     2                                      4                                                2                                                                  3                           |
   \                                                  (-1 + x*cot(x))                       (-1 + x*cot(x))                        (-1 + x*cot(x))                                  (-1 + x*cot(x))                                                    (-1 + x*cot(x))                            /
−2(3x(x(cot⁡2(x)+1)−cot⁡(x))3(xcot⁡(x)−1)4+6x(x(cot⁡2(x)+1)−cot⁡(x))(−x(cot⁡2(x)+1)cot⁡(x)+cot⁡2(x)+1)(xcot⁡(x)−1)3+x(cot⁡2(x)+1)(x(cot⁡2(x)+1)+2xcot⁡2(x)−3cot⁡(x))(xcot⁡(x)−1)2+3(x(cot⁡2(x)+1)−cot⁡(x))2(xcot⁡(x)−1)3+(cot⁡2(x)+1)2+2(cot⁡2(x)+1)cot⁡2(x)+3(−x(cot⁡2(x)+1)cot⁡(x)+cot⁡2(x)+1)(xcot⁡(x)−1)2)- 2 \left(\frac{3 x \left(x \left(\cot^{2}{\left(x \right)} + 1\right) - \cot{\left(x \right)}\right)^{3}}{\left(x \cot{\left(x \right)} - 1\right)^{4}} + \frac{6 x \left(x \left(\cot^{2}{\left(x \right)} + 1\right) - \cot{\left(x \right)}\right) \left(- x \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 1\right)}{\left(x \cot{\left(x \right)} - 1\right)^{3}} + \frac{x \left(\cot^{2}{\left(x \right)} + 1\right) \left(x \left(\cot^{2}{\left(x \right)} + 1\right) + 2 x \cot^{2}{\left(x \right)} - 3 \cot{\left(x \right)}\right)}{\left(x \cot{\left(x \right)} - 1\right)^{2}} + \frac{3 \left(x \left(\cot^{2}{\left(x \right)} + 1\right) - \cot{\left(x \right)}\right)^{2}}{\left(x \cot{\left(x \right)} - 1\right)^{3}} + \left(\cot^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} + \frac{3 \left(- x \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 1\right)}{\left(x \cot{\left(x \right)} - 1\right)^{2}}\right)
Simplificar
Gráfico
Derivada de y=ctgx+x/(1-xctgx)
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