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(tan(x)-cot(x))/(tan(x)+cot(x))
Derivada de la función/ (tan(x)-cot(x))/(tan(x)+cot(x))

Derivada de (tan(x)-cot(x))/(tan(x)+cot(x))

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

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Solución

Ha introducido [src] 
tan(x) - cot(x)
---------------
tan(x) + cot(x)
tan⁡(x)−cot⁡(x)tan⁡(x)+cot⁡(x)\frac{\tan{\left(x \right)} - \cot{\left(x \right)}}{\tan{\left(x \right)} + \cot{\left(x \right)}} 
(tan(x) - cot(x))/(tan(x) + 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)=tan⁡(x)−cot⁡(x)f{\left(x \right)} = \tan{\left(x \right)} - \cot{\left(x \right)} y g(x)=tan⁡(x)+cot⁡(x)g{\left(x \right)} = \tan{\left(x \right)} + \cot{\left(x \right)}.

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

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

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

        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)}}

        Entonces, como resultado: 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)}}

      2. ddxtan⁡(x)=1cos⁡2(x)\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}

      Como resultado de: sin⁡2(x)+cos⁡2(x)cos⁡2(x)+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)}} + \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\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. diferenciamos tan⁡(x)+cot⁡(x)\tan{\left(x \right)} + \cot{\left(x \right)} miembro por miembro:

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

      2. ddxtan⁡(x)=1cos⁡2(x)\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}

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

    Ahora aplicamos la regla de la derivada de una divesión:

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

  2. Simplificamos:

    4tan⁡(x)−4cos⁡3(x)sin⁡(x)\frac{4}{\tan{\left(x \right)}} - \frac{4 \cos^{3}{\left(x \right)}}{\sin{\left(x \right)}}

Respuesta:

4tan⁡(x)−4cos⁡3(x)sin⁡(x)\frac{4}{\tan{\left(x \right)}} - \frac{4 \cos^{3}{\left(x \right)}}{\sin{\left(x \right)}}

Gráfica
02468-8-6-4-2-10105-5
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
       2         2      /   2         2   \                  
2 + cot (x) + tan (x)   \cot (x) - tan (x)/*(tan(x) - cot(x))
--------------------- + -------------------------------------
   tan(x) + cot(x)                                 2         
                                  (tan(x) + cot(x))
(tan⁡(x)−cot⁡(x))(−tan⁡2(x)+cot⁡2(x))(tan⁡(x)+cot⁡(x))2+tan⁡2(x)+cot⁡2(x)+2tan⁡(x)+cot⁡(x)\frac{\left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) \left(- \tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)}\right)}{\left(\tan{\left(x \right)} + \cot{\left(x \right)}\right)^{2}} + \frac{\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2}{\tan{\left(x \right)} + \cot{\left(x \right)}}
Simplificar
Segunda derivada [src] 
  /                                                                                                               /                                                                 2\\
  |                                                                                                               |                                              /   2         2   \ ||
  |                                                                                                               |/       2   \          /       2   \          \tan (x) - cot (x)/ ||
  |                                              /   2         2   \ /       2         2   \   (-cot(x) + tan(x))*|\1 + cot (x)/*cot(x) + \1 + tan (x)/*tan(x) - --------------------||
  |/       2   \          /       2   \          \tan (x) - cot (x)/*\2 + cot (x) + tan (x)/                      \                                                cot(x) + tan(x)   /|
2*|\1 + tan (x)/*tan(x) - \1 + cot (x)/*cot(x) - ------------------------------------------- - ---------------------------------------------------------------------------------------|
  \                                                            cot(x) + tan(x)                                                     cot(x) + tan(x)                                    /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                    cot(x) + tan(x)
2(−(tan⁡(x)−cot⁡(x))((tan⁡2(x)+1)tan⁡(x)+(cot⁡2(x)+1)cot⁡(x)−(tan⁡2(x)−cot⁡2(x))2tan⁡(x)+cot⁡(x))tan⁡(x)+cot⁡(x)+(tan⁡2(x)+1)tan⁡(x)−(cot⁡2(x)+1)cot⁡(x)−(tan⁡2(x)−cot⁡2(x))(tan⁡2(x)+cot⁡2(x)+2)tan⁡(x)+cot⁡(x))tan⁡(x)+cot⁡(x)\frac{2 \left(- \frac{\left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \frac{\left(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right)^{2}}{\tan{\left(x \right)} + \cot{\left(x \right)}}\right)}{\tan{\left(x \right)} + \cot{\left(x \right)}} + \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \frac{\left(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)}{\tan{\left(x \right)} + \cot{\left(x \right)}}\right)}{\tan{\left(x \right)} + \cot{\left(x \right)}}
Simplificar
Tercera derivada [src] 
  /                                                                                                         /                                                       3                                                                                                                          \                                                                                                                                                                       \
  |                                                                                                         |             2                2     /   2         2   \                                                          /   2         2   \ //       2   \          /       2   \       \|                                                                                                   /                                                                 2\|
  |                                                                                                         |/       2   \    /       2   \    3*\tan (x) - cot (x)/         2    /       2   \        2    /       2   \   6*\tan (x) - cot (x)/*\\1 + cot (x)/*cot(x) + \1 + tan (x)/*tan(x)/|                                                                                                   |                                              /   2         2   \ ||
  |                                                                                      (-cot(x) + tan(x))*|\1 + cot (x)/  - \1 + tan (x)/  - ---------------------- - 2*tan (x)*\1 + tan (x)/ + 2*cot (x)*\1 + cot (x)/ + -------------------------------------------------------------------|                                                                           /       2         2   \ |/       2   \          /       2   \          \tan (x) - cot (x)/ ||
  |             2                2                                                                          |                                                     2                                                                                   cot(x) + tan(x)                          |     /   2         2   \ //       2   \          /       2   \       \   3*\2 + cot (x) + tan (x)/*|\1 + cot (x)/*cot(x) + \1 + tan (x)/*tan(x) - --------------------||
  |/       2   \    /       2   \         2    /       2   \        2    /       2   \                      \                                    (cot(x) + tan(x))                                                                                                                             /   3*\tan (x) - cot (x)/*\\1 + tan (x)/*tan(x) - \1 + cot (x)/*cot(x)/                             \                                                cot(x) + tan(x)   /|
2*|\1 + cot (x)/  + \1 + tan (x)/  + 2*cot (x)*\1 + cot (x)/ + 2*tan (x)*\1 + tan (x)/ + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------|
  \                                                                                                                                                                                  cot(x) + tan(x)                                                                                                                         cot(x) + tan(x)                                                                    cot(x) + tan(x)                                        /
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                            cot(x) + tan(x)
2(−3((tan⁡2(x)+1)tan⁡(x)−(cot⁡2(x)+1)cot⁡(x))(tan⁡2(x)−cot⁡2(x))tan⁡(x)+cot⁡(x)+(tan⁡(x)−cot⁡(x))(6((tan⁡2(x)+1)tan⁡(x)+(cot⁡2(x)+1)cot⁡(x))(tan⁡2(x)−cot⁡2(x))tan⁡(x)+cot⁡(x)−(tan⁡2(x)+1)2−2(tan⁡2(x)+1)tan⁡2(x)+(cot⁡2(x)+1)2+2(cot⁡2(x)+1)cot⁡2(x)−3(tan⁡2(x)−cot⁡2(x))3(tan⁡(x)+cot⁡(x))2)tan⁡(x)+cot⁡(x)+(tan⁡2(x)+1)2+2(tan⁡2(x)+1)tan⁡2(x)+(cot⁡2(x)+1)2+2(cot⁡2(x)+1)cot⁡2(x)−3((tan⁡2(x)+1)tan⁡(x)+(cot⁡2(x)+1)cot⁡(x)−(tan⁡2(x)−cot⁡2(x))2tan⁡(x)+cot⁡(x))(tan⁡2(x)+cot⁡2(x)+2)tan⁡(x)+cot⁡(x))tan⁡(x)+cot⁡(x)\frac{2 \left(- \frac{3 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right)}{\tan{\left(x \right)} + \cot{\left(x \right)}} + \frac{\left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) \left(\frac{6 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right)}{\tan{\left(x \right)} + \cot{\left(x \right)}} - \left(\tan^{2}{\left(x \right)} + 1\right)^{2} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \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(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right)^{3}}{\left(\tan{\left(x \right)} + \cot{\left(x \right)}\right)^{2}}\right)}{\tan{\left(x \right)} + \cot{\left(x \right)}} + \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \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(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \frac{\left(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right)^{2}}{\tan{\left(x \right)} + \cot{\left(x \right)}}\right) \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)}{\tan{\left(x \right)} + \cot{\left(x \right)}}\right)}{\tan{\left(x \right)} + \cot{\left(x \right)}}
Simplificar
Gráfico
Derivada de (tan(x)-cot(x))/(tan(x)+cot(x))
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