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

Derivada de cot(x)/tan(x)

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

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

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

          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)}}

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

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

    Ahora aplicamos la regla de la derivada de una divesión:

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

  2. Simplificamos:

    −2cos⁡(x)sin⁡3(x)- \frac{2 \cos{\left(x \right)}}{\sin^{3}{\left(x \right)}}

Respuesta:

−2cos⁡(x)sin⁡3(x)- \frac{2 \cos{\left(x \right)}}{\sin^{3}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-50000005000000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
        2      /        2   \       
-1 - cot (x)   \-1 - tan (x)/*cot(x)
------------ + ---------------------
   tan(x)                2          
                      tan (x)
(−tan⁡2(x)−1)cot⁡(x)tan⁡2(x)+−cot⁡2(x)−1tan⁡(x)\frac{\left(- \tan^{2}{\left(x \right)} - 1\right) \cot{\left(x \right)}}{\tan^{2}{\left(x \right)}} + \frac{- \cot^{2}{\left(x \right)} - 1}{\tan{\left(x \right)}}
Simplificar
Segunda derivada [src] 
  /                       /       2   \ /       2   \                 /            2   \       \
  |/       2   \          \1 + cot (x)/*\1 + tan (x)/   /       2   \ |     1 + tan (x)|       |
2*|\1 + cot (x)/*cot(x) + --------------------------- + \1 + tan (x)/*|-1 + -----------|*cot(x)|
  |                                  tan(x)                           |          2     |       |
  \                                                                   \       tan (x)  /       /
------------------------------------------------------------------------------------------------
                                             tan(x)
2((tan⁡2(x)+1tan⁡2(x)−1)(tan⁡2(x)+1)cot⁡(x)+(tan⁡2(x)+1)(cot⁡2(x)+1)tan⁡(x)+(cot⁡2(x)+1)cot⁡(x))tan⁡(x)\frac{2 \left(\left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right)}{\tan{\left(x \right)}}
Simplificar
Tercera derivada [src] 
   /                                                                                                                             /            2   \                                       \
   |                                                                                                 /       2   \ /       2   \ |     1 + tan (x)|                                       |
   |/                               2                  3\                                          3*\1 + cot (x)/*\1 + tan (x)/*|-1 + -----------|                                       |
   ||                  /       2   \      /       2   \ |          /       2   \ /         2   \                                 |          2     |     /       2   \ /       2   \       |
   ||         2      5*\1 + tan (x)/    3*\1 + tan (x)/ |          \1 + cot (x)/*\1 + 3*cot (x)/                                 \       tan (x)  /   3*\1 + cot (x)/*\1 + tan (x)/*cot(x)|
-2*||2 + 2*tan (x) - ---------------- + ----------------|*cot(x) + ----------------------------- + ------------------------------------------------ + ------------------------------------|
   ||                       2                  4        |                      tan(x)                                   tan(x)                                         2                  |
   \\                    tan (x)            tan (x)     /                                                                                                           tan (x)               /
−2(3(tan⁡2(x)+1tan⁡2(x)−1)(tan⁡2(x)+1)(cot⁡2(x)+1)tan⁡(x)+3(tan⁡2(x)+1)(cot⁡2(x)+1)cot⁡(x)tan⁡2(x)+(cot⁡2(x)+1)(3cot⁡2(x)+1)tan⁡(x)+(3(tan⁡2(x)+1)3tan⁡4(x)−5(tan⁡2(x)+1)2tan⁡2(x)+2tan⁡2(x)+2)cot⁡(x))- 2 \left(\frac{3 \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{\tan^{2}{\left(x \right)}} + \frac{\left(\cot^{2}{\left(x \right)} + 1\right) \left(3 \cot^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + \left(\frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{3}}{\tan^{4}{\left(x \right)}} - \frac{5 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} + 2 \tan^{2}{\left(x \right)} + 2\right) \cot{\left(x \right)}\right)
Simplificar
Gráfico
Derivada de cot(x)/tan(x)
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