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y=tg^4cbrt*x
Derivada de la función/ y=tg^4/ y=tg^4cbrt*x

Derivada de y=tg^4cbrt*x

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

Ha introducido [src] 
   4    3 ___
tan (x)*\/ x
x3tan⁡4(x)\sqrt[3]{x} \tan^{4}{\left(x \right)} 
tan(x)^4*x^(1/3)
Solución detallada

  1. Se aplica la regla de la derivada de una multiplicación:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=tan⁡4(x)f{\left(x \right)} = \tan^{4}{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Sustituimos u=tan⁡(x)u = \tan{\left(x \right)}.

    2. Según el principio, aplicamos: u4u^{4} tenemos 4u34 u^{3}

    3. 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:

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

    g(x)=x3g{\left(x \right)} = \sqrt[3]{x}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Según el principio, aplicamos: x3\sqrt[3]{x} tenemos 13x23\frac{1}{3 x^{\frac{2}{3}}}

    Como resultado de: 4x3(sin⁡2(x)+cos⁡2(x))tan⁡3(x)cos⁡2(x)+tan⁡4(x)3x23\frac{4 \sqrt[3]{x} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\tan^{4}{\left(x \right)}}{3 x^{\frac{2}{3}}}

  2. Simplificamos:

    12xsin⁡3(x)cos⁡5(x)+tan⁡4(x)3x23\frac{\frac{12 x \sin^{3}{\left(x \right)}}{\cos^{5}{\left(x \right)}} + \tan^{4}{\left(x \right)}}{3 x^{\frac{2}{3}}}

Respuesta:

12xsin⁡3(x)cos⁡5(x)+tan⁡4(x)3x23\frac{\frac{12 x \sin^{3}{\left(x \right)}}{\cos^{5}{\left(x \right)}} + \tan^{4}{\left(x \right)}}{3 x^{\frac{2}{3}}}

Gráfica
02468-8-6-4-2-1010-200000000100000000
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Primera derivada [src] 
   4                                   
tan (x)   3 ___    3    /         2   \
------- + \/ x *tan (x)*\4 + 4*tan (x)/
    2/3                                
 3*x
x3(4tan⁡2(x)+4)tan⁡3(x)+tan⁡4(x)3x23\sqrt[3]{x} \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)} + \frac{\tan^{4}{\left(x \right)}}{3 x^{\frac{2}{3}}}
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Segunda derivada [src] 
          /     2                                                /       2   \       \
     2    |  tan (x)     3 ___ /       2   \ /         2   \   4*\1 + tan (x)/*tan(x)|
2*tan (x)*|- ------- + 2*\/ x *\1 + tan (x)/*\3 + 5*tan (x)/ + ----------------------|
          |      5/3                                                      2/3        |
          \   9*x                                                      3*x           /
2(2x3(tan⁡2(x)+1)(5tan⁡2(x)+3)+4(tan⁡2(x)+1)tan⁡(x)3x23−tan⁡2(x)9x53)tan⁡2(x)2 \left(2 \sqrt[3]{x} \left(\tan^{2}{\left(x \right)} + 1\right) \left(5 \tan^{2}{\left(x \right)} + 3\right) + \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{3 x^{\frac{2}{3}}} - \frac{\tan^{2}{\left(x \right)}}{9 x^{\frac{5}{3}}}\right) \tan^{2}{\left(x \right)}
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Tercera derivada [src] 
  /     3                            /                           2                           \        2    /       2   \     /       2   \ /         2   \       \       
  |5*tan (x)     3 ___ /       2   \ |     4        /       2   \          2    /       2   \|   4*tan (x)*\1 + tan (x)/   2*\1 + tan (x)/*\3 + 5*tan (x)/*tan(x)|       
2*|--------- + 4*\/ x *\1 + tan (x)/*\2*tan (x) + 3*\1 + tan (x)/  + 10*tan (x)*\1 + tan (x)// - ----------------------- + --------------------------------------|*tan(x)
  |     8/3                                                                                                  5/3                             2/3                 |       
  \ 27*x                                                                                                  3*x                               x                    /
2(4x3(tan⁡2(x)+1)(3(tan⁡2(x)+1)2+10(tan⁡2(x)+1)tan⁡2(x)+2tan⁡4(x))+2(tan⁡2(x)+1)(5tan⁡2(x)+3)tan⁡(x)x23−4(tan⁡2(x)+1)tan⁡2(x)3x53+5tan⁡3(x)27x83)tan⁡(x)2 \left(4 \sqrt[3]{x} \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 10 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 2 \tan^{4}{\left(x \right)}\right) + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(5 \tan^{2}{\left(x \right)} + 3\right) \tan{\left(x \right)}}{x^{\frac{2}{3}}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)}}{3 x^{\frac{5}{3}}} + \frac{5 \tan^{3}{\left(x \right)}}{27 x^{\frac{8}{3}}}\right) \tan{\left(x \right)}
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Gráfico
Derivada de y=tg^4cbrt*x
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