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y=tan^4(xsinx)
Derivada de la función/ y=tan/ xsinx/ y=tan^4(xsinx)

Derivada de y=tan^4(xsinx)

Función f() - derivada -er orden en el punto
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Ha introducido [src] 
   4          
tan (x*sin(x))
tan⁡4(xsin⁡(x))\tan^{4}{\left(x \sin{\left(x \right)} \right)} 
tan(x*sin(x))^4
Solución detallada

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

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

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddxtan⁡(xsin⁡(x))\frac{d}{d x} \tan{\left(x \sin{\left(x \right)} \right)}:

    1. Reescribimos las funciones para diferenciar:

      tan⁡(xsin⁡(x))=sin⁡(xsin⁡(x))cos⁡(xsin⁡(x))\tan{\left(x \sin{\left(x \right)} \right)} = \frac{\sin{\left(x \sin{\left(x \right)} \right)}}{\cos{\left(x \sin{\left(x \right)} \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⁡(xsin⁡(x))f{\left(x \right)} = \sin{\left(x \sin{\left(x \right)} \right)} y g(x)=cos⁡(xsin⁡(x))g{\left(x \right)} = \cos{\left(x \sin{\left(x \right)} \right)}.

      Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

      2. La derivada del seno es igual al coseno:

        ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

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

        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)=xf{\left(x \right)} = x; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

          1. Según el principio, aplicamos: xx tenemos 11

          g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\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: xcos⁡(x)+sin⁡(x)x \cos{\left(x \right)} + \sin{\left(x \right)}

        Como resultado de la secuencia de reglas:

        (xcos⁡(x)+sin⁡(x))cos⁡(xsin⁡(x))\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \sin{\left(x \right)} \right)}

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

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

      2. La derivada del coseno es igual a menos el seno:

        dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

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

        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)=xf{\left(x \right)} = x; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

          1. Según el principio, aplicamos: xx tenemos 11

          g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\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: xcos⁡(x)+sin⁡(x)x \cos{\left(x \right)} + \sin{\left(x \right)}

        Como resultado de la secuencia de reglas:

        −(xcos⁡(x)+sin⁡(x))sin⁡(xsin⁡(x))- \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}

      Ahora aplicamos la regla de la derivada de una divesión:

      (xcos⁡(x)+sin⁡(x))sin⁡2(xsin⁡(x))+(xcos⁡(x)+sin⁡(x))cos⁡2(xsin⁡(x))cos⁡2(xsin⁡(x))\frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{2}{\left(x \sin{\left(x \right)} \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos^{2}{\left(x \sin{\left(x \right)} \right)}}{\cos^{2}{\left(x \sin{\left(x \right)} \right)}}

    Como resultado de la secuencia de reglas:

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

  4. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-200000000200000000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
     3           /       2          \                    
4*tan (x*sin(x))*\1 + tan (x*sin(x))/*(x*cos(x) + sin(x))
4(xcos⁡(x)+sin⁡(x))(tan⁡2(xsin⁡(x))+1)tan⁡3(xsin⁡(x))4 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\tan^{2}{\left(x \sin{\left(x \right)} \right)} + 1\right) \tan^{3}{\left(x \sin{\left(x \right)} \right)}
Simplificar
Segunda derivada [src] 
     2           /       2          \ /                                                             2    2                                  2 /       2          \\
4*tan (x*sin(x))*\1 + tan (x*sin(x))/*\-(-2*cos(x) + x*sin(x))*tan(x*sin(x)) + 2*(x*cos(x) + sin(x)) *tan (x*sin(x)) + 3*(x*cos(x) + sin(x)) *\1 + tan (x*sin(x))//
4(tan⁡2(xsin⁡(x))+1)(−(xsin⁡(x)−2cos⁡(x))tan⁡(xsin⁡(x))+3(xcos⁡(x)+sin⁡(x))2(tan⁡2(xsin⁡(x))+1)+2(xcos⁡(x)+sin⁡(x))2tan⁡2(xsin⁡(x)))tan⁡2(xsin⁡(x))4 \left(\tan^{2}{\left(x \sin{\left(x \right)} \right)} + 1\right) \left(- \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \tan{\left(x \sin{\left(x \right)} \right)} + 3 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \left(\tan^{2}{\left(x \sin{\left(x \right)} \right)} + 1\right) + 2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \tan^{2}{\left(x \sin{\left(x \right)} \right)}\right) \tan^{2}{\left(x \sin{\left(x \right)} \right)}
Simplificar
Tercera derivada [src] 
                       /                                                                                                       2                                                                                                                                                                                                                                   \              
  /       2          \ |     2                                                        3    4               /       2          \                     3        3                                                                              3    2           /       2          \     /       2          \                                                         |              
4*\1 + tan (x*sin(x))/*\- tan (x*sin(x))*(3*sin(x) + x*cos(x)) + 4*(x*cos(x) + sin(x)) *tan (x*sin(x)) + 6*\1 + tan (x*sin(x))/ *(x*cos(x) + sin(x))  - 6*tan (x*sin(x))*(-2*cos(x) + x*sin(x))*(x*cos(x) + sin(x)) + 20*(x*cos(x) + sin(x)) *tan (x*sin(x))*\1 + tan (x*sin(x))/ - 9*\1 + tan (x*sin(x))/*(-2*cos(x) + x*sin(x))*(x*cos(x) + sin(x))*tan(x*sin(x))/*tan(x*sin(x))
4(tan⁡2(xsin⁡(x))+1)(−9(xsin⁡(x)−2cos⁡(x))(xcos⁡(x)+sin⁡(x))(tan⁡2(xsin⁡(x))+1)tan⁡(xsin⁡(x))−6(xsin⁡(x)−2cos⁡(x))(xcos⁡(x)+sin⁡(x))tan⁡3(xsin⁡(x))+6(xcos⁡(x)+sin⁡(x))3(tan⁡2(xsin⁡(x))+1)2+20(xcos⁡(x)+sin⁡(x))3(tan⁡2(xsin⁡(x))+1)tan⁡2(xsin⁡(x))+4(xcos⁡(x)+sin⁡(x))3tan⁡4(xsin⁡(x))−(xcos⁡(x)+3sin⁡(x))tan⁡2(xsin⁡(x)))tan⁡(xsin⁡(x))4 \left(\tan^{2}{\left(x \sin{\left(x \right)} \right)} + 1\right) \left(- 9 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\tan^{2}{\left(x \sin{\left(x \right)} \right)} + 1\right) \tan{\left(x \sin{\left(x \right)} \right)} - 6 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \tan^{3}{\left(x \sin{\left(x \right)} \right)} + 6 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{3} \left(\tan^{2}{\left(x \sin{\left(x \right)} \right)} + 1\right)^{2} + 20 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{3} \left(\tan^{2}{\left(x \sin{\left(x \right)} \right)} + 1\right) \tan^{2}{\left(x \sin{\left(x \right)} \right)} + 4 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{3} \tan^{4}{\left(x \sin{\left(x \right)} \right)} - \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \tan^{2}{\left(x \sin{\left(x \right)} \right)}\right) \tan{\left(x \sin{\left(x \right)} \right)}
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Derivada de y=tan^4(xsinx)
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