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y=(tgx)^4*(x)^5
Derivada de la función/ (tgx)/ y=(tgx)^4*(x)^5

Derivada de y=(tgx)^4*(x)^5

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

Ha introducido [src] 
   4     5
tan (x)*x
x5tan⁡4(x)x^{5} \tan^{4}{\left(x \right)} 
tan(x)^4*x^5
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)=x5g{\left(x \right)} = x^{5}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Según el principio, aplicamos: x5x^{5} tenemos 5x45 x^{4}

    Como resultado de: 4x5(sin⁡2(x)+cos⁡2(x))tan⁡3(x)cos⁡2(x)+5x4tan⁡4(x)\frac{4 x^{5} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5 x^{4} \tan^{4}{\left(x \right)}

  2. Simplificamos:

    x4(4xsin⁡3(x)cos⁡5(x)+5tan⁡4(x))x^{4} \left(\frac{4 x \sin^{3}{\left(x \right)}}{\cos^{5}{\left(x \right)}} + 5 \tan^{4}{\left(x \right)}\right)

Respuesta:

x4(4xsin⁡3(x)cos⁡5(x)+5tan⁡4(x))x^{4} \left(\frac{4 x \sin^{3}{\left(x \right)}}{\cos^{5}{\left(x \right)}} + 5 \tan^{4}{\left(x \right)}\right)

Gráfica
02468-8-6-4-2-1010-250000000000250000000000
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Primera derivada [src] 
   4    4       5    3    /         2   \
5*x *tan (x) + x *tan (x)*\4 + 4*tan (x)/
x5(4tan⁡2(x)+4)tan⁡3(x)+5x4tan⁡4(x)x^{5} \left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)} + 5 x^{4} \tan^{4}{\left(x \right)}
Simplificar
Segunda derivada [src] 
   3    2    /     2       2 /       2   \ /         2   \        /       2   \       \
4*x *tan (x)*\5*tan (x) + x *\1 + tan (x)/*\3 + 5*tan (x)/ + 10*x*\1 + tan (x)/*tan(x)/
4x3(x2(tan⁡2(x)+1)(5tan⁡2(x)+3)+10x(tan⁡2(x)+1)tan⁡(x)+5tan⁡2(x))tan⁡2(x)4 x^{3} \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \left(5 \tan^{2}{\left(x \right)} + 3\right) + 10 x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 5 \tan^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}
Simplificar
Tercera derivada [src] 
     /                                /                           2                           \                                                                          \       
   2 |      3         3 /       2   \ |     4        /       2   \          2    /       2   \|           2    /       2   \       2 /       2   \ /         2   \       |       
4*x *\15*tan (x) + 2*x *\1 + tan (x)/*\2*tan (x) + 3*\1 + tan (x)/  + 10*tan (x)*\1 + tan (x)// + 60*x*tan (x)*\1 + tan (x)/ + 15*x *\1 + tan (x)/*\3 + 5*tan (x)/*tan(x)/*tan(x)
4x2(2x3(tan⁡2(x)+1)(3(tan⁡2(x)+1)2+10(tan⁡2(x)+1)tan⁡2(x)+2tan⁡4(x))+15x2(tan⁡2(x)+1)(5tan⁡2(x)+3)tan⁡(x)+60x(tan⁡2(x)+1)tan⁡2(x)+15tan⁡3(x))tan⁡(x)4 x^{2} \left(2 x^{3} \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) + 15 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \left(5 \tan^{2}{\left(x \right)} + 3\right) \tan{\left(x \right)} + 60 x \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 15 \tan^{3}{\left(x \right)}\right) \tan{\left(x \right)}
Simplificar
Gráfico
Derivada de y=(tgx)^4*(x)^5
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