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(tan(x))/(sin(x)-cos(x))
Derivada de la función/ sin(x)/ tan(x)/ cos(x)/ (tan(x))/(sin(x)-cos(x))

Derivada de (tan(x))/(sin(x)-cos(x))

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

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Solución

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

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

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

    1. diferenciamos sin⁡(x)−cos⁡(x)\sin{\left(x \right)} - \cos{\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. 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)}

        Entonces, como resultado: sin⁡(x)\sin{\left(x \right)}

      2. La derivada del seno es igual al coseno:

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

      Como resultado de: sin⁡(x)+cos⁡(x)\sin{\left(x \right)} + \cos{\left(x \right)}

    Ahora aplicamos la regla de la derivada de una divesión:

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

  2. Simplificamos:

    2(sin⁡(x)sin⁡(x+π4)cos⁡(x)+cos⁡(x+π4))(sin⁡(2x)−1)cos⁡2(x)\frac{\sqrt{2} \left(\sin{\left(x \right)} \sin{\left(x + \frac{\pi}{4} \right)} \cos{\left(x \right)} + \cos{\left(x + \frac{\pi}{4} \right)}\right)}{\left(\sin{\left(2 x \right)} - 1\right) \cos^{2}{\left(x \right)}}

Respuesta:

2(sin⁡(x)sin⁡(x+π4)cos⁡(x)+cos⁡(x+π4))(sin⁡(2x)−1)cos⁡2(x)\frac{\sqrt{2} \left(\sin{\left(x \right)} \sin{\left(x + \frac{\pi}{4} \right)} \cos{\left(x \right)} + \cos{\left(x + \frac{\pi}{4} \right)}\right)}{\left(\sin{\left(2 x \right)} - 1\right) \cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-50005000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
         2                                 
  1 + tan (x)     (-cos(x) - sin(x))*tan(x)
--------------- + -------------------------
sin(x) - cos(x)                        2   
                      (sin(x) - cos(x))
(−sin⁡(x)−cos⁡(x))tan⁡(x)(sin⁡(x)−cos⁡(x))2+tan⁡2(x)+1sin⁡(x)−cos⁡(x)\frac{\left(- \sin{\left(x \right)} - \cos{\left(x \right)}\right) \tan{\left(x \right)}}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}} + \frac{\tan^{2}{\left(x \right)} + 1}{\sin{\left(x \right)} - \cos{\left(x \right)}}
Simplificar
Segunda derivada [src] 
/                       2\                                     /       2   \                  
|    2*(cos(x) + sin(x)) |            /       2   \          2*\1 + tan (x)/*(cos(x) + sin(x))
|1 + --------------------|*tan(x) + 2*\1 + tan (x)/*tan(x) - ---------------------------------
|                      2 |                                            -cos(x) + sin(x)        
\    (-cos(x) + sin(x))  /                                                                    
----------------------------------------------------------------------------------------------
                                       -cos(x) + sin(x)
(1+2(sin⁡(x)+cos⁡(x))2(sin⁡(x)−cos⁡(x))2)tan⁡(x)+2(tan⁡2(x)+1)tan⁡(x)−2(sin⁡(x)+cos⁡(x))(tan⁡2(x)+1)sin⁡(x)−cos⁡(x)sin⁡(x)−cos⁡(x)\frac{\left(1 + \frac{2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}\right) \tan{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}}{\sin{\left(x \right)} - \cos{\left(x \right)}}
Simplificar
Tercera derivada [src] 
                                                                               /                       2\                                                                    
                                                                               |    6*(cos(x) + sin(x)) |                                                                    
                                                                               |5 + --------------------|*(cos(x) + sin(x))*tan(x)                                           
                                                  /                       2\   |                      2 |                              /       2   \                         
  /       2   \ /         2   \     /       2   \ |    2*(cos(x) + sin(x)) |   \    (-cos(x) + sin(x))  /                            6*\1 + tan (x)/*(cos(x) + sin(x))*tan(x)
2*\1 + tan (x)/*\1 + 3*tan (x)/ + 3*\1 + tan (x)/*|1 + --------------------| - --------------------------------------------------- - ----------------------------------------
                                                  |                      2 |                     -cos(x) + sin(x)                                -cos(x) + sin(x)            
                                                  \    (-cos(x) + sin(x))  /                                                                                                 
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                               -cos(x) + sin(x)
3(1+2(sin⁡(x)+cos⁡(x))2(sin⁡(x)−cos⁡(x))2)(tan⁡2(x)+1)−(5+6(sin⁡(x)+cos⁡(x))2(sin⁡(x)−cos⁡(x))2)(sin⁡(x)+cos⁡(x))tan⁡(x)sin⁡(x)−cos⁡(x)+2(tan⁡2(x)+1)(3tan⁡2(x)+1)−6(sin⁡(x)+cos⁡(x))(tan⁡2(x)+1)tan⁡(x)sin⁡(x)−cos⁡(x)sin⁡(x)−cos⁡(x)\frac{3 \left(1 + \frac{2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{\left(5 + \frac{6 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \tan{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) - \frac{6 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}}}{\sin{\left(x \right)} - \cos{\left(x \right)}}
Simplificar
Gráfico
Derivada de (tan(x))/(sin(x)-cos(x))
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