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y=x*tg^3(x^2-1)x*exp(-x)
Derivada de la función/ x^2-1/ x*exp/ exp(-x)/ y=x*tg^3(x^2-1)/ y=x*tg^3(x^2-1)x*exp(-x)

Derivada de y=x*tg^3(x^2-1)x*exp(-x)

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

Ha introducido [src] 
     3/ 2    \    -x
x*tan \x  - 1/*x*e
xxtan⁡3(x2−1)e−xx x \tan^{3}{\left(x^{2} - 1 \right)} e^{- x} 
((x*tan(x^2 - 1)^3)*x)*exp(-x)
Solución detallada

  1. 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)=x2tan⁡3(x2−1)f{\left(x \right)} = x^{2} \tan^{3}{\left(x^{2} - 1 \right)} y g(x)=exg{\left(x \right)} = e^{x}.

    Para calcular ddxf(x)\frac{d}{d x} f{\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)=x2f{\left(x \right)} = x^{2}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Según el principio, aplicamos: x2x^{2} tenemos 2x2 x

      g(x)=tan⁡3(x2−1)g{\left(x \right)} = \tan^{3}{\left(x^{2} - 1 \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Sustituimos u=tan⁡(x2−1)u = \tan{\left(x^{2} - 1 \right)}.

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

      3. Luego se aplica una cadena de reglas. Multiplicamos por ddxtan⁡(x2−1)\frac{d}{d x} \tan{\left(x^{2} - 1 \right)}:

        1. Reescribimos las funciones para diferenciar:

          tan⁡(x2−1)=sin⁡(x2−1)cos⁡(x2−1)\tan{\left(x^{2} - 1 \right)} = \frac{\sin{\left(x^{2} - 1 \right)}}{\cos{\left(x^{2} - 1 \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⁡(x2−1)f{\left(x \right)} = \sin{\left(x^{2} - 1 \right)} y g(x)=cos⁡(x2−1)g{\left(x \right)} = \cos{\left(x^{2} - 1 \right)}.

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

          1. Sustituimos u=x2−1u = x^{2} - 1.

          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 ddx(x2−1)\frac{d}{d x} \left(x^{2} - 1\right):

            1. diferenciamos x2−1x^{2} - 1 miembro por miembro:

              1. Según el principio, aplicamos: x2x^{2} tenemos 2x2 x

              2. La derivada de una constante −1-1 es igual a cero.

              Como resultado de: 2x2 x

            Como resultado de la secuencia de reglas:

            2xcos⁡(x2−1)2 x \cos{\left(x^{2} - 1 \right)}

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

          1. Sustituimos u=x2−1u = x^{2} - 1.

          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 ddx(x2−1)\frac{d}{d x} \left(x^{2} - 1\right):

            1. diferenciamos x2−1x^{2} - 1 miembro por miembro:

              1. Según el principio, aplicamos: x2x^{2} tenemos 2x2 x

              2. La derivada de una constante −1-1 es igual a cero.

              Como resultado de: 2x2 x

            Como resultado de la secuencia de reglas:

            −2xsin⁡(x2−1)- 2 x \sin{\left(x^{2} - 1 \right)}

          Ahora aplicamos la regla de la derivada de una divesión:

          2xsin⁡2(x2−1)+2xcos⁡2(x2−1)cos⁡2(x2−1)\frac{2 x \sin^{2}{\left(x^{2} - 1 \right)} + 2 x \cos^{2}{\left(x^{2} - 1 \right)}}{\cos^{2}{\left(x^{2} - 1 \right)}}

        Como resultado de la secuencia de reglas:

        3(2xsin⁡2(x2−1)+2xcos⁡2(x2−1))tan⁡2(x2−1)cos⁡2(x2−1)\frac{3 \left(2 x \sin^{2}{\left(x^{2} - 1 \right)} + 2 x \cos^{2}{\left(x^{2} - 1 \right)}\right) \tan^{2}{\left(x^{2} - 1 \right)}}{\cos^{2}{\left(x^{2} - 1 \right)}}

      Como resultado de: 3x2(2xsin⁡2(x2−1)+2xcos⁡2(x2−1))tan⁡2(x2−1)cos⁡2(x2−1)+2xtan⁡3(x2−1)\frac{3 x^{2} \left(2 x \sin^{2}{\left(x^{2} - 1 \right)} + 2 x \cos^{2}{\left(x^{2} - 1 \right)}\right) \tan^{2}{\left(x^{2} - 1 \right)}}{\cos^{2}{\left(x^{2} - 1 \right)}} + 2 x \tan^{3}{\left(x^{2} - 1 \right)}

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

    1. Derivado exe^{x} es.

    Ahora aplicamos la regla de la derivada de una divesión:

    (−x2extan⁡3(x2−1)+(3x2(2xsin⁡2(x2−1)+2xcos⁡2(x2−1))tan⁡2(x2−1)cos⁡2(x2−1)+2xtan⁡3(x2−1))ex)e−2x\left(- x^{2} e^{x} \tan^{3}{\left(x^{2} - 1 \right)} + \left(\frac{3 x^{2} \left(2 x \sin^{2}{\left(x^{2} - 1 \right)} + 2 x \cos^{2}{\left(x^{2} - 1 \right)}\right) \tan^{2}{\left(x^{2} - 1 \right)}}{\cos^{2}{\left(x^{2} - 1 \right)}} + 2 x \tan^{3}{\left(x^{2} - 1 \right)}\right) e^{x}\right) e^{- 2 x}

  2. Simplificamos:

    6x3e−xtan⁡2(x2−1)cos⁡2(x2−1)−x2e−xtan⁡3(x2−1)+2xe−xtan⁡3(x2−1)\frac{6 x^{3} e^{- x} \tan^{2}{\left(x^{2} - 1 \right)}}{\cos^{2}{\left(x^{2} - 1 \right)}} - x^{2} e^{- x} \tan^{3}{\left(x^{2} - 1 \right)} + 2 x e^{- x} \tan^{3}{\left(x^{2} - 1 \right)}

Respuesta:

6x3e−xtan⁡2(x2−1)cos⁡2(x2−1)−x2e−xtan⁡3(x2−1)+2xe−xtan⁡3(x2−1)\frac{6 x^{3} e^{- x} \tan^{2}{\left(x^{2} - 1 \right)}}{\cos^{2}{\left(x^{2} - 1 \right)}} - x^{2} e^{- x} \tan^{3}{\left(x^{2} - 1 \right)} + 2 x e^{- x} \tan^{3}{\left(x^{2} - 1 \right)}

Gráfica
02468-8-6-4-2-1010-40000000000002000000000000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
/     3/ 2    \     /   3/ 2    \      2    2/ 2    \ /       2/ 2    \\\\  -x    2    3/ 2    \  -x
\x*tan \x  - 1/ + x*\tan \x  - 1/ + 6*x *tan \x  - 1/*\1 + tan \x  - 1////*e   - x *tan \x  - 1/*e
−x2e−xtan⁡3(x2−1)+(x(6x2(tan⁡2(x2−1)+1)tan⁡2(x2−1)+tan⁡3(x2−1))+xtan⁡3(x2−1))e−x- x^{2} e^{- x} \tan^{3}{\left(x^{2} - 1 \right)} + \left(x \left(6 x^{2} \left(\tan^{2}{\left(x^{2} - 1 \right)} + 1\right) \tan^{2}{\left(x^{2} - 1 \right)} + \tan^{3}{\left(x^{2} - 1 \right)}\right) + x \tan^{3}{\left(x^{2} - 1 \right)}\right) e^{- x}
Simplificar
Segunda derivada [src] 
/     2/      2\    2    2/      2\       /   2 /       2/      2\\      /      2\\    /      2\      2 /       2/      2\\ /     /      2\      2    2/      2\      2 /       2/      2\\\       2 /       2/      2\\    /      2\\  -x    /      2\
\2*tan \-1 + x / + x *tan \-1 + x / - 4*x*\3*x *\1 + tan \-1 + x // + tan\-1 + x //*tan\-1 + x / + 6*x *\1 + tan \-1 + x //*\3*tan\-1 + x / + 4*x *tan \-1 + x / + 4*x *\1 + tan \-1 + x /// + 12*x *\1 + tan \-1 + x //*tan\-1 + x //*e  *tan\-1 + x /
(6x2(tan⁡2(x2−1)+1)(4x2(tan⁡2(x2−1)+1)+4x2tan⁡2(x2−1)+3tan⁡(x2−1))+12x2(tan⁡2(x2−1)+1)tan⁡(x2−1)+x2tan⁡2(x2−1)−4x(3x2(tan⁡2(x2−1)+1)+tan⁡(x2−1))tan⁡(x2−1)+2tan⁡2(x2−1))e−xtan⁡(x2−1)\left(6 x^{2} \left(\tan^{2}{\left(x^{2} - 1 \right)} + 1\right) \left(4 x^{2} \left(\tan^{2}{\left(x^{2} - 1 \right)} + 1\right) + 4 x^{2} \tan^{2}{\left(x^{2} - 1 \right)} + 3 \tan{\left(x^{2} - 1 \right)}\right) + 12 x^{2} \left(\tan^{2}{\left(x^{2} - 1 \right)} + 1\right) \tan{\left(x^{2} - 1 \right)} + x^{2} \tan^{2}{\left(x^{2} - 1 \right)} - 4 x \left(3 x^{2} \left(\tan^{2}{\left(x^{2} - 1 \right)} + 1\right) + \tan{\left(x^{2} - 1 \right)}\right) \tan{\left(x^{2} - 1 \right)} + 2 \tan^{2}{\left(x^{2} - 1 \right)}\right) e^{- x} \tan{\left(x^{2} - 1 \right)}
Simplificar
Tercera derivada [src] 
/                                                                                                                                                                                                                                                                                /                                                                                                                                                                       /                                          2                                                                                                    \\\    
|   2    3/      2\     /   2/      2\      2 /       2/      2\\ /     /      2\      2    2/      2\      2 /       2/      2\\\      2 /       2/      2\\    /      2\\    /      2\          2/      2\ /   2 /       2/      2\\      /      2\\       /       2/      2\\ |  /     /      2\      2    2/      2\      2 /       2/      2\\\    /      2\     /   2    2/      2\      2 /       2/      2\\      /      2\\    /      2\      2 |     3/      2\      2 /       2/      2\\      /       2/      2\\    /      2\      2    4/      2\       2    2/      2\ /       2/      2\\|||  -x
\- x *tan \-1 + x / - 6*\tan \-1 + x / + 3*x *\1 + tan \-1 + x //*\3*tan\-1 + x / + 4*x *tan \-1 + x / + 4*x *\1 + tan \-1 + x /// + 6*x *\1 + tan \-1 + x //*tan\-1 + x //*tan\-1 + x / + 6*x*tan \-1 + x /*\3*x *\1 + tan \-1 + x // + tan\-1 + x // + 6*x*\1 + tan \-1 + x //*\3*\3*tan\-1 + x / + 4*x *tan \-1 + x / + 4*x *\1 + tan \-1 + x ///*tan\-1 + x / + 3*\4*x *tan \-1 + x / + 4*x *\1 + tan \-1 + x // + tan\-1 + x //*tan\-1 + x / + 4*x *\3*tan \-1 + x / + 2*x *\1 + tan \-1 + x //  + 3*\1 + tan \-1 + x //*tan\-1 + x / + 4*x *tan \-1 + x / + 14*x *tan \-1 + x /*\1 + tan \-1 + x /////*e
(−x2tan⁡3(x2−1)+6x(3x2(tan⁡2(x2−1)+1)+tan⁡(x2−1))tan⁡2(x2−1)+6x(tan⁡2(x2−1)+1)(4x2(2x2(tan⁡2(x2−1)+1)2+14x2(tan⁡2(x2−1)+1)tan⁡2(x2−1)+4x2tan⁡4(x2−1)+3(tan⁡2(x2−1)+1)tan⁡(x2−1)+3tan⁡3(x2−1))+3(4x2(tan⁡2(x2−1)+1)+4x2tan⁡2(x2−1)+tan⁡(x2−1))tan⁡(x2−1)+3(4x2(tan⁡2(x2−1)+1)+4x2tan⁡2(x2−1)+3tan⁡(x2−1))tan⁡(x2−1))−6(3x2(tan⁡2(x2−1)+1)(4x2(tan⁡2(x2−1)+1)+4x2tan⁡2(x2−1)+3tan⁡(x2−1))+6x2(tan⁡2(x2−1)+1)tan⁡(x2−1)+tan⁡2(x2−1))tan⁡(x2−1))e−x\left(- x^{2} \tan^{3}{\left(x^{2} - 1 \right)} + 6 x \left(3 x^{2} \left(\tan^{2}{\left(x^{2} - 1 \right)} + 1\right) + \tan{\left(x^{2} - 1 \right)}\right) \tan^{2}{\left(x^{2} - 1 \right)} + 6 x \left(\tan^{2}{\left(x^{2} - 1 \right)} + 1\right) \left(4 x^{2} \left(2 x^{2} \left(\tan^{2}{\left(x^{2} - 1 \right)} + 1\right)^{2} + 14 x^{2} \left(\tan^{2}{\left(x^{2} - 1 \right)} + 1\right) \tan^{2}{\left(x^{2} - 1 \right)} + 4 x^{2} \tan^{4}{\left(x^{2} - 1 \right)} + 3 \left(\tan^{2}{\left(x^{2} - 1 \right)} + 1\right) \tan{\left(x^{2} - 1 \right)} + 3 \tan^{3}{\left(x^{2} - 1 \right)}\right) + 3 \left(4 x^{2} \left(\tan^{2}{\left(x^{2} - 1 \right)} + 1\right) + 4 x^{2} \tan^{2}{\left(x^{2} - 1 \right)} + \tan{\left(x^{2} - 1 \right)}\right) \tan{\left(x^{2} - 1 \right)} + 3 \left(4 x^{2} \left(\tan^{2}{\left(x^{2} - 1 \right)} + 1\right) + 4 x^{2} \tan^{2}{\left(x^{2} - 1 \right)} + 3 \tan{\left(x^{2} - 1 \right)}\right) \tan{\left(x^{2} - 1 \right)}\right) - 6 \left(3 x^{2} \left(\tan^{2}{\left(x^{2} - 1 \right)} + 1\right) \left(4 x^{2} \left(\tan^{2}{\left(x^{2} - 1 \right)} + 1\right) + 4 x^{2} \tan^{2}{\left(x^{2} - 1 \right)} + 3 \tan{\left(x^{2} - 1 \right)}\right) + 6 x^{2} \left(\tan^{2}{\left(x^{2} - 1 \right)} + 1\right) \tan{\left(x^{2} - 1 \right)} + \tan^{2}{\left(x^{2} - 1 \right)}\right) \tan{\left(x^{2} - 1 \right)}\right) e^{- x}
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Gráfico
Derivada de y=x*tg^3(x^2-1)x*exp(-x)
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