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tan(x)/x^(2/3)
Derivada de la función/ tan(x)/ x^(2/3)/ tan(x)/x^(2/3)

Derivada de tan(x)/x^(2/3)

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

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Solución

Ha introducido [src] 
tan(x)
------
  2/3 
 x
tan⁡(x)x23\frac{\tan{\left(x \right)}}{x^{\frac{2}{3}}} 
tan(x)/x^(2/3)
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)=x23g{\left(x \right)} = x^{\frac{2}{3}}.

    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. Según el principio, aplicamos: x23x^{\frac{2}{3}} tenemos 23x3\frac{2}{3 \sqrt[3]{x}}

    Ahora aplicamos la regla de la derivada de una divesión:

    x23(sin⁡2(x)+cos⁡2(x))cos⁡2(x)−2tan⁡(x)3x3x43\frac{\frac{x^{\frac{2}{3}} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \frac{2 \tan{\left(x \right)}}{3 \sqrt[3]{x}}}{x^{\frac{4}{3}}}

  2. Simplificamos:

    x−sin⁡(2x)3x53cos⁡2(x)\frac{x - \frac{\sin{\left(2 x \right)}}{3}}{x^{\frac{5}{3}} \cos^{2}{\left(x \right)}}

Respuesta:

x−sin⁡(2x)3x53cos⁡2(x)\frac{x - \frac{\sin{\left(2 x \right)}}{3}}{x^{\frac{5}{3}} \cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-500500
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
       2              
1 + tan (x)   2*tan(x)
----------- - --------
     2/3          5/3 
    x          3*x
tan⁡2(x)+1x23−2tan⁡(x)3x53\frac{\tan^{2}{\left(x \right)} + 1}{x^{\frac{2}{3}}} - \frac{2 \tan{\left(x \right)}}{3 x^{\frac{5}{3}}}
Simplificar
Segunda derivada [src] 
  /                         /       2   \           \
  |/       2   \          2*\1 + tan (x)/   5*tan(x)|
2*|\1 + tan (x)/*tan(x) - --------------- + --------|
  |                             3*x              2  |
  \                                           9*x   /
-----------------------------------------------------
                          2/3                        
                         x
2((tan⁡2(x)+1)tan⁡(x)−2(tan⁡2(x)+1)3x+5tan⁡(x)9x2)x23\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{3 x} + \frac{5 \tan{\left(x \right)}}{9 x^{2}}\right)}{x^{\frac{2}{3}}}
Simplificar
Tercera derivada [src] 
  /                                              /       2   \     /       2   \       \
  |/       2   \ /         2   \   40*tan(x)   5*\1 + tan (x)/   2*\1 + tan (x)/*tan(x)|
2*|\1 + tan (x)/*\1 + 3*tan (x)/ - --------- + --------------- - ----------------------|
  |                                      3              2                  x           |
  \                                  27*x            3*x                               /
----------------------------------------------------------------------------------------
                                           2/3                                          
                                          x
2((tan⁡2(x)+1)(3tan⁡2(x)+1)−2(tan⁡2(x)+1)tan⁡(x)x+5(tan⁡2(x)+1)3x2−40tan⁡(x)27x3)x23\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} + \frac{5 \left(\tan^{2}{\left(x \right)} + 1\right)}{3 x^{2}} - \frac{40 \tan{\left(x \right)}}{27 x^{3}}\right)}{x^{\frac{2}{3}}}
Simplificar
Gráfico
Derivada de tan(x)/x^(2/3)
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