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x^4/tan(x)
Derivada de la función/ tan(x)/ x^4/tan(x)

Derivada de x^4/tan(x)

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

Ha introducido [src] 
   4  
  x   
------
tan(x)
x4tan⁡(x)\frac{x^{4}}{\tan{\left(x \right)}} 
x^4/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)=x4f{\left(x \right)} = x^{4} 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. Según el principio, aplicamos: x4x^{4} tenemos 4x34 x^{3}

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

    Ahora aplicamos la regla de la derivada de una divesión:

    −x4(sin⁡2(x)+cos⁡2(x))cos⁡2(x)+4x3tan⁡(x)tan⁡2(x)\frac{- \frac{x^{4} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + 4 x^{3} \tan{\left(x \right)}}{\tan^{2}{\left(x \right)}}

  2. Simplificamos:

    x3(−xsin⁡2(x)+4tan⁡(x))x^{3} \left(- \frac{x}{\sin^{2}{\left(x \right)}} + \frac{4}{\tan{\left(x \right)}}\right)

Respuesta:

x3(−xsin⁡2(x)+4tan⁡(x))x^{3} \left(- \frac{x}{\sin^{2}{\left(x \right)}} + \frac{4}{\tan{\left(x \right)}}\right)

Gráfica
02468-8-6-4-2-1010-1000000010000000
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Primera derivada [src] 
    3     4 /        2   \
 4*x     x *\-1 - tan (x)/
------ + -----------------
tan(x)           2        
              tan (x)
x4(−tan⁡2(x)−1)tan⁡2(x)+4x3tan⁡(x)\frac{x^{4} \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} + \frac{4 x^{3}}{\tan{\left(x \right)}}
Simplificar
Segunda derivada [src] 
     /                     /            2   \       /       2   \\
   2 |     2 /       2   \ |     1 + tan (x)|   4*x*\1 + tan (x)/|
2*x *|6 + x *\1 + tan (x)/*|-1 + -----------| - -----------------|
     |                     |          2     |         tan(x)     |
     \                     \       tan (x)  /                    /
------------------------------------------------------------------
                              tan(x)
2x2(x2(tan⁡2(x)+1tan⁡2(x)−1)(tan⁡2(x)+1)−4x(tan⁡2(x)+1)tan⁡(x)+6)tan⁡(x)\frac{2 x^{2} \left(x^{2} \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{4 x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + 6\right)}{\tan{\left(x \right)}}
Simplificar
Tercera derivada [src] 
    /                                                                                                             /            2   \\
    |                                                                                             2 /       2   \ |     1 + tan (x)||
    |            /                               2                  3\                        12*x *\1 + tan (x)/*|-1 + -----------||
    |            |                  /       2   \      /       2   \ |        /       2   \                       |          2     ||
    |  12      3 |         2      5*\1 + tan (x)/    3*\1 + tan (x)/ |   18*x*\1 + tan (x)/                       \       tan (x)  /|
2*x*|------ - x *|2 + 2*tan (x) - ---------------- + ----------------| - ------------------ + --------------------------------------|
    |tan(x)      |                       2                  4        |           2                            tan(x)                |
    \            \                    tan (x)            tan (x)     /        tan (x)                                               /
2x(−x3(3(tan⁡2(x)+1)3tan⁡4(x)−5(tan⁡2(x)+1)2tan⁡2(x)+2tan⁡2(x)+2)+12x2(tan⁡2(x)+1tan⁡2(x)−1)(tan⁡2(x)+1)tan⁡(x)−18x(tan⁡2(x)+1)tan⁡2(x)+12tan⁡(x))2 x \left(- x^{3} \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) + \frac{12 x^{2} \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} - \frac{18 x \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan^{2}{\left(x \right)}} + \frac{12}{\tan{\left(x \right)}}\right)
Simplificar
Gráfico
Derivada de x^4/tan(x)
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