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

Derivada de y=tg^4sqrt*x

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

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Solución

Ha introducido [src] 
   4      ___
tan (x)*\/ x
xtan⁡4(x)\sqrt{x} \tan^{4}{\left(x \right)} 
tan(x)^4*sqrt(x)
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)=xg{\left(x \right)} = \sqrt{x}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Según el principio, aplicamos: x\sqrt{x} tenemos 12x\frac{1}{2 \sqrt{x}}

    Como resultado de: 4x(sin⁡2(x)+cos⁡2(x))tan⁡3(x)cos⁡2(x)+tan⁡4(x)2x\frac{4 \sqrt{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)}}{2 \sqrt{x}}

  2. Simplificamos:

    8xsin⁡3(x)cos⁡5(x)+tan⁡4(x)2x\frac{\frac{8 x \sin^{3}{\left(x \right)}}{\cos^{5}{\left(x \right)}} + \tan^{4}{\left(x \right)}}{2 \sqrt{x}}

Respuesta:

8xsin⁡3(x)cos⁡5(x)+tan⁡4(x)2x\frac{\frac{8 x \sin^{3}{\left(x \right)}}{\cos^{5}{\left(x \right)}} + \tan^{4}{\left(x \right)}}{2 \sqrt{x}}

Gráfica
02468-8-6-4-2-1010-200000000200000000
Trazar el gráfico f(x)
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Primera derivada [src] 
   4                                   
tan (x)     ___    3    /         2   \
------- + \/ x *tan (x)*\4 + 4*tan (x)/
    ___                                
2*\/ x
x(4tan⁡2(x)+4)tan⁡3(x)+tan⁡4(x)2x\sqrt{x} \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)} + \frac{\tan^{4}{\left(x \right)}}{2 \sqrt{x}}
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Segunda derivada [src] 
        /     2                                                /       2   \       \
   2    |  tan (x)       ___ /       2   \ /         2   \   4*\1 + tan (x)/*tan(x)|
tan (x)*|- ------- + 4*\/ x *\1 + tan (x)/*\3 + 5*tan (x)/ + ----------------------|
        |      3/2                                                     ___         |
        \   4*x                                                      \/ x          /
(4x(tan⁡2(x)+1)(5tan⁡2(x)+3)+4(tan⁡2(x)+1)tan⁡(x)x−tan⁡2(x)4x32)tan⁡2(x)\left(4 \sqrt{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)}}{\sqrt{x}} - \frac{\tan^{2}{\left(x \right)}}{4 x^{\frac{3}{2}}}\right) \tan^{2}{\left(x \right)}
Simplificar
Tercera derivada [src] 
/     3           2    /       2   \                         /                           2                           \     /       2   \ /         2   \       \       
|3*tan (x)   3*tan (x)*\1 + tan (x)/       ___ /       2   \ |     4        /       2   \          2    /       2   \|   6*\1 + tan (x)/*\3 + 5*tan (x)/*tan(x)|       
|--------- - ----------------------- + 8*\/ x *\1 + tan (x)/*\2*tan (x) + 3*\1 + tan (x)/  + 10*tan (x)*\1 + tan (x)// + --------------------------------------|*tan(x)
|     5/2               3/2                                                                                                                ___                 |       
\  8*x                 x                                                                                                                 \/ x                  /
(8x(tan⁡2(x)+1)(3(tan⁡2(x)+1)2+10(tan⁡2(x)+1)tan⁡2(x)+2tan⁡4(x))+6(tan⁡2(x)+1)(5tan⁡2(x)+3)tan⁡(x)x−3(tan⁡2(x)+1)tan⁡2(x)x32+3tan⁡3(x)8x52)tan⁡(x)\left(8 \sqrt{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{6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(5 \tan^{2}{\left(x \right)} + 3\right) \tan{\left(x \right)}}{\sqrt{x}} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)}}{x^{\frac{3}{2}}} + \frac{3 \tan^{3}{\left(x \right)}}{8 x^{\frac{5}{2}}}\right) \tan{\left(x \right)}
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Gráfico
Derivada de y=tg^4sqrt*x
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