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(1+tan(x))/(1-tan(x))
Derivada de la función/ tan(x)/ (1+tan(x))/(1-tan(x))

Derivada de (1+tan(x))/(1-tan(x))

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

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Solución

Ha introducido [src] 
1 + tan(x)
----------
1 - tan(x)
tan⁡(x)+11−tan⁡(x)\frac{\tan{\left(x \right)} + 1}{1 - \tan{\left(x \right)}} 
(1 + tan(x))/(1 - 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)=tan⁡(x)+1f{\left(x \right)} = \tan{\left(x \right)} + 1 y g(x)=1−tan⁡(x)g{\left(x \right)} = 1 - \tan{\left(x \right)}.

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

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

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

      2. Reescribimos las funciones para diferenciar:

        tan⁡(x)=sin⁡(x)cos⁡(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

      3. 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: 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)}}

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

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

        Entonces, como resultado: −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: −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)}}

    Ahora aplicamos la regla de la derivada de una divesión:

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

  2. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-500010000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
       2      /       2   \             
1 + tan (x)   \1 + tan (x)/*(1 + tan(x))
----------- + --------------------------
 1 - tan(x)                     2       
                    (1 - tan(x))
tan⁡2(x)+11−tan⁡(x)+(tan⁡(x)+1)(tan⁡2(x)+1)(1−tan⁡(x))2\frac{\tan^{2}{\left(x \right)} + 1}{1 - \tan{\left(x \right)}} + \frac{\left(\tan{\left(x \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(1 - \tan{\left(x \right)}\right)^{2}}
Simplificar
Segunda derivada [src] 
                /                                     /         2            \\
                |                                     |  1 + tan (x)         ||
                |                 2      (1 + tan(x))*|- ----------- + tan(x)||
  /       2   \ |          1 + tan (x)                \  -1 + tan(x)         /|
2*\1 + tan (x)/*|-tan(x) + ----------- + -------------------------------------|
                \          -1 + tan(x)                -1 + tan(x)             /
-------------------------------------------------------------------------------
                                  -1 + tan(x)
2(tan⁡2(x)+1)(−tan⁡(x)+(tan⁡(x)+1)(tan⁡(x)−tan⁡2(x)+1tan⁡(x)−1)tan⁡(x)−1+tan⁡2(x)+1tan⁡(x)−1)tan⁡(x)−1\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(- \tan{\left(x \right)} + \frac{\left(\tan{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)} - 1}\right)}{\tan{\left(x \right)} - 1} + \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)} - 1}\right)}{\tan{\left(x \right)} - 1}
Simplificar
Tercera derivada [src] 
                /                              /                               2                         \                                                                    \
                |                              |                  /       2   \      /       2   \       |                                                                    |
                |                              |         2      3*\1 + tan (x)/    6*\1 + tan (x)/*tan(x)|                   /         2            \                         |
                |                 (1 + tan(x))*|1 + 3*tan (x) + ---------------- - ----------------------|     /       2   \ |  1 + tan (x)         |                         |
                |                              |                              2         -1 + tan(x)      |   3*\1 + tan (x)/*|- ----------- + tan(x)|     /       2   \       |
  /       2   \ |          2                   \                 (-1 + tan(x))                           /                   \  -1 + tan(x)         /   3*\1 + tan (x)/*tan(x)|
2*\1 + tan (x)/*|-1 - 3*tan (x) + ------------------------------------------------------------------------ + ---------------------------------------- + ----------------------|
                \                                               -1 + tan(x)                                                -1 + tan(x)                       -1 + tan(x)      /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                  -1 + tan(x)
2(tan⁡2(x)+1)(−3tan⁡2(x)−1+(tan⁡(x)+1)(3tan⁡2(x)+1−6(tan⁡2(x)+1)tan⁡(x)tan⁡(x)−1+3(tan⁡2(x)+1)2(tan⁡(x)−1)2)tan⁡(x)−1+3(tan⁡(x)−tan⁡2(x)+1tan⁡(x)−1)(tan⁡2(x)+1)tan⁡(x)−1+3(tan⁡2(x)+1)tan⁡(x)tan⁡(x)−1)tan⁡(x)−1\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(- 3 \tan^{2}{\left(x \right)} - 1 + \frac{\left(\tan{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1 - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{\tan{\left(x \right)} - 1} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\left(\tan{\left(x \right)} - 1\right)^{2}}\right)}{\tan{\left(x \right)} - 1} + \frac{3 \left(\tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)} - 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)} - 1} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{\tan{\left(x \right)} - 1}\right)}{\tan{\left(x \right)} - 1}
Simplificar
Gráfico
Derivada de (1+tan(x))/(1-tan(x))
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