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y=sin^3x*tg^52x
Derivada de la función/ sin^3/ y=sin/ y=sin^3x*tg^52x

Derivada de y=sin^3x*tg^52x

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

Ha introducido [src] 
   3       52   
sin (x)*tan  (x)
sin⁡3(x)tan⁡52(x)\sin^{3}{\left(x \right)} \tan^{52}{\left(x \right)} 
sin(x)^3*tan(x)^52
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)=sin⁡3(x)f{\left(x \right)} = \sin^{3}{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

    2. Según el principio, aplicamos: u3u^{3} tenemos 3u23 u^{2}

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddxsin⁡(x)\frac{d}{d x} \sin{\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)}

      Como resultado de la secuencia de reglas:

      3sin⁡2(x)cos⁡(x)3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}

    g(x)=tan⁡52(x)g{\left(x \right)} = \tan^{52}{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

    2. Según el principio, aplicamos: u52u^{52} tenemos 52u5152 u^{51}

    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:

      52(sin⁡2(x)+cos⁡2(x))tan⁡51(x)cos⁡2(x)\frac{52 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{51}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

    Como resultado de: 52(sin⁡2(x)+cos⁡2(x))sin⁡3(x)tan⁡51(x)cos⁡2(x)+3sin⁡2(x)cos⁡(x)tan⁡52(x)\frac{52 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin^{3}{\left(x \right)} \tan^{51}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} \tan^{52}{\left(x \right)}

  2. Simplificamos:

    (3+52cos⁡2(x))sin⁡54(x)cos⁡51(x)\frac{\left(3 + \frac{52}{\cos^{2}{\left(x \right)}}\right) \sin^{54}{\left(x \right)}}{\cos^{51}{\left(x \right)}}

Respuesta:

(3+52cos⁡2(x))sin⁡54(x)cos⁡51(x)\frac{\left(3 + \frac{52}{\cos^{2}{\left(x \right)}}\right) \sin^{54}{\left(x \right)}}{\cos^{51}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-5e7810e78
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Primera derivada [src] 
   3       51    /           2   \        2       52          
sin (x)*tan  (x)*\52 + 52*tan (x)/ + 3*sin (x)*tan  (x)*cos(x)
(52tan⁡2(x)+52)sin⁡3(x)tan⁡51(x)+3sin⁡2(x)cos⁡(x)tan⁡52(x)\left(52 \tan^{2}{\left(x \right)} + 52\right) \sin^{3}{\left(x \right)} \tan^{51}{\left(x \right)} + 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} \tan^{52}{\left(x \right)}
Simplificar
Segunda derivada [src] 
   50    /       2    /   2           2   \         2    /       2   \ /           2   \       /       2   \                     \       
tan  (x)*\- 3*tan (x)*\sin (x) - 2*cos (x)/ + 52*sin (x)*\1 + tan (x)/*\51 + 53*tan (x)/ + 312*\1 + tan (x)/*cos(x)*sin(x)*tan(x)/*sin(x)
(−3(sin⁡2(x)−2cos⁡2(x))tan⁡2(x)+52(tan⁡2(x)+1)(53tan⁡2(x)+51)sin⁡2(x)+312(tan⁡2(x)+1)sin⁡(x)cos⁡(x)tan⁡(x))sin⁡(x)tan⁡50(x)\left(- 3 \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)} + 52 \left(\tan^{2}{\left(x \right)} + 1\right) \left(53 \tan^{2}{\left(x \right)} + 51\right) \sin^{2}{\left(x \right)} + 312 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)}\right) \sin{\left(x \right)} \tan^{50}{\left(x \right)}
Simplificar
Tercera derivada [src] 
         /                                                                         /                              2                            \                                                                                                                     \
   49    |       3    /       2           2   \                 3    /       2   \ |     4           /       2   \           2    /       2   \|          2    /       2   \ /   2           2   \                 2    /       2   \ /           2   \              |
tan  (x)*\- 3*tan (x)*\- 2*cos (x) + 7*sin (x)/*cos(x) + 104*sin (x)*\1 + tan (x)/*\2*tan (x) + 1275*\1 + tan (x)/  + 154*tan (x)*\1 + tan (x)// - 468*tan (x)*\1 + tan (x)/*\sin (x) - 2*cos (x)/*sin(x) + 468*sin (x)*\1 + tan (x)/*\51 + 53*tan (x)/*cos(x)*tan(x)/
(−468(sin⁡2(x)−2cos⁡2(x))(tan⁡2(x)+1)sin⁡(x)tan⁡2(x)−3(7sin⁡2(x)−2cos⁡2(x))cos⁡(x)tan⁡3(x)+468(tan⁡2(x)+1)(53tan⁡2(x)+51)sin⁡2(x)cos⁡(x)tan⁡(x)+104(tan⁡2(x)+1)(1275(tan⁡2(x)+1)2+154(tan⁡2(x)+1)tan⁡2(x)+2tan⁡4(x))sin⁡3(x))tan⁡49(x)\left(- 468 \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan^{2}{\left(x \right)} - 3 \left(7 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)} \tan^{3}{\left(x \right)} + 468 \left(\tan^{2}{\left(x \right)} + 1\right) \left(53 \tan^{2}{\left(x \right)} + 51\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)} + 104 \left(\tan^{2}{\left(x \right)} + 1\right) \left(1275 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 154 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 2 \tan^{4}{\left(x \right)}\right) \sin^{3}{\left(x \right)}\right) \tan^{49}{\left(x \right)}
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Gráfico
Derivada de y=sin^3x*tg^52x
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