Sr Examen

Otras calculadoras

x*exp(-x)1/5tg^5+2/3tg^3x+tgx
Derivada de la función/ x*exp(-x)1/5tg^5+2/3tg^3x+tgx

Derivada de x*exp(-x)1/5tg^5+2/3tg^3x+tgx

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
   -x                3            
x*e      5      2*tan (x)         
-----*tan (x) + --------- + tan(x)
  5                 3
(xe−x5tan⁡5(x)+2tan⁡3(x)3)+tan⁡(x)\left(\frac{x e^{- x}}{5} \tan^{5}{\left(x \right)} + \frac{2 \tan^{3}{\left(x \right)}}{3}\right) + \tan{\left(x \right)} 
((x*exp(-x))/5)*tan(x)^5 + 2*tan(x)^3/3 + tan(x)
Solución detallada

  1. diferenciamos (xe−x5tan⁡5(x)+2tan⁡3(x)3)+tan⁡(x)\left(\frac{x e^{- x}}{5} \tan^{5}{\left(x \right)} + \frac{2 \tan^{3}{\left(x \right)}}{3}\right) + \tan{\left(x \right)} miembro por miembro:

    1. diferenciamos xe−x5tan⁡5(x)+2tan⁡3(x)3\frac{x e^{- x}}{5} \tan^{5}{\left(x \right)} + \frac{2 \tan^{3}{\left(x \right)}}{3} miembro por miembro:

      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)=xtan⁡5(x)f{\left(x \right)} = x \tan^{5}{\left(x \right)} y g(x)=5exg{\left(x \right)} = 5 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)=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)=tan⁡5(x)g{\left(x \right)} = \tan^{5}{\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: u5u^{5} tenemos 5u45 u^{4}

          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:

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

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

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

        1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.

          1. Derivado exe^{x} es.

          Entonces, como resultado: 5ex5 e^{x}

        Ahora aplicamos la regla de la derivada de una divesión:

        (−5xextan⁡5(x)+5(5x(sin⁡2(x)+cos⁡2(x))tan⁡4(x)cos⁡2(x)+tan⁡5(x))ex)e−2x25\frac{\left(- 5 x e^{x} \tan^{5}{\left(x \right)} + 5 \left(\frac{5 x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{4}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \tan^{5}{\left(x \right)}\right) e^{x}\right) e^{- 2 x}}{25}

      2. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.

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

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

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

          1. ddxtan⁡(x)=1cos⁡2(x)\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}

          Como resultado de la secuencia de reglas:

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

        Entonces, como resultado: 2(sin⁡2(x)+cos⁡2(x))tan⁡2(x)cos⁡2(x)\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

      Como resultado de: (−5xextan⁡5(x)+5(5x(sin⁡2(x)+cos⁡2(x))tan⁡4(x)cos⁡2(x)+tan⁡5(x))ex)e−2x25+2(sin⁡2(x)+cos⁡2(x))tan⁡2(x)cos⁡2(x)\frac{\left(- 5 x e^{x} \tan^{5}{\left(x \right)} + 5 \left(\frac{5 x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{4}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \tan^{5}{\left(x \right)}\right) e^{x}\right) e^{- 2 x}}{25} + \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

    2. ddxtan⁡(x)=1cos⁡2(x)\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}

    Como resultado de: (−5xextan⁡5(x)+5(5x(sin⁡2(x)+cos⁡2(x))tan⁡4(x)cos⁡2(x)+tan⁡5(x))ex)e−2x25+2(sin⁡2(x)+cos⁡2(x))tan⁡2(x)cos⁡2(x)+sin⁡2(x)+cos⁡2(x)cos⁡2(x)\frac{\left(- 5 x e^{x} \tan^{5}{\left(x \right)} + 5 \left(\frac{5 x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{4}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \tan^{5}{\left(x \right)}\right) e^{x}\right) e^{- 2 x}}{25} + \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

  2. Simplificamos:

    (5(2tan⁡2(x)+1)ex+(−xsin⁡(2x)+10x+sin⁡(2x))tan⁡4(x)2)e−x5cos⁡2(x)\frac{\left(5 \left(2 \tan^{2}{\left(x \right)} + 1\right) e^{x} + \frac{\left(- x \sin{\left(2 x \right)} + 10 x + \sin{\left(2 x \right)}\right) \tan^{4}{\left(x \right)}}{2}\right) e^{- x}}{5 \cos^{2}{\left(x \right)}}

Respuesta:

(5(2tan⁡2(x)+1)ex+(−xsin⁡(2x)+10x+sin⁡(2x))tan⁡4(x)2)e−x5cos⁡2(x)\frac{\left(5 \left(2 \tan^{2}{\left(x \right)} + 1\right) e^{x} + \frac{\left(- x \sin{\left(2 x \right)} + 10 x + \sin{\left(2 x \right)}\right) \tan^{4}{\left(x \right)}}{2}\right) e^{- x}}{5 \cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-500000000000500000000000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
                      / -x      -x\        2    /         2   \        4    /         2   \  -x
       2         5    |e     x*e  |   2*tan (x)*\3 + 3*tan (x)/   x*tan (x)*\5 + 5*tan (x)/*e  
1 + tan (x) + tan (x)*|--- - -----| + ------------------------- + -----------------------------
                      \ 5      5  /               3                             5
x(5tan⁡2(x)+5)e−xtan⁡4(x)5+(−xe−x5+e−x5)tan⁡5(x)+2(3tan⁡2(x)+3)tan⁡2(x)3+tan⁡2(x)+1\frac{x \left(5 \tan^{2}{\left(x \right)} + 5\right) e^{- x} \tan^{4}{\left(x \right)}}{5} + \left(- \frac{x e^{- x}}{5} + \frac{e^{- x}}{5}\right) \tan^{5}{\left(x \right)} + \frac{2 \left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)}}{3} + \tan^{2}{\left(x \right)} + 1
Simplificar
Segunda derivada [src] 
/                               2                                                            4              -x                                                                                                                       2            \       
|         2        /       2   \         2    /       2   \      3    /       2   \  -x   tan (x)*(-2 + x)*e          3    /       2   \  -x      3    /       2   \           -x          4    /       2   \  -x       /       2   \     2     -x|       
|2 + 2*tan (x) + 4*\1 + tan (x)/  + 4*tan (x)*\1 + tan (x)/ + tan (x)*\1 + tan (x)/*e   + -------------------- - x*tan (x)*\1 + tan (x)/*e   - tan (x)*\1 + tan (x)/*(-1 + x)*e   + 2*x*tan (x)*\1 + tan (x)/*e   + 4*x*\1 + tan (x)/ *tan (x)*e  |*tan(x)
\                                                                                                  5                                                                                                                                              /
(4x(tan⁡2(x)+1)2e−xtan⁡2(x)+2x(tan⁡2(x)+1)e−xtan⁡4(x)−x(tan⁡2(x)+1)e−xtan⁡3(x)+(x−2)e−xtan⁡4(x)5−(x−1)(tan⁡2(x)+1)e−xtan⁡3(x)+4(tan⁡2(x)+1)2+4(tan⁡2(x)+1)tan⁡2(x)+(tan⁡2(x)+1)e−xtan⁡3(x)+2tan⁡2(x)+2)tan⁡(x)\left(4 x \left(\tan^{2}{\left(x \right)} + 1\right)^{2} e^{- x} \tan^{2}{\left(x \right)} + 2 x \left(\tan^{2}{\left(x \right)} + 1\right) e^{- x} \tan^{4}{\left(x \right)} - x \left(\tan^{2}{\left(x \right)} + 1\right) e^{- x} \tan^{3}{\left(x \right)} + \frac{\left(x - 2\right) e^{- x} \tan^{4}{\left(x \right)}}{5} - \left(x - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) e^{- x} \tan^{3}{\left(x \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) e^{- x} \tan^{3}{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}
Simplificar
Tercera derivada [src] 
               2                  3                                                                       2                                                                                      2                  5              -x                                                  2                                                              2                                                                                                                                                        3                                 2            
  /       2   \      /       2   \         2    /       2   \        4    /       2   \      /       2   \     2           4    /       2   \  -x        5    /       2   \  -x     /       2   \     3     -x   tan (x)*(-3 + x)*e          4    /       2   \  -x       /       2   \     3     -x          5    /       2   \  -x     /       2   \     3              -x        5    /       2   \           -x        4    /       2   \           -x          6    /       2   \  -x        /       2   \     2     -x        /       2   \     4     -x
2*\1 + tan (x)/  + 4*\1 + tan (x)/  + 4*tan (x)*\1 + tan (x)/ + 8*tan (x)*\1 + tan (x)/ + 28*\1 + tan (x)/ *tan (x) - 2*tan (x)*\1 + tan (x)/*e   + 4*tan (x)*\1 + tan (x)/*e   + 8*\1 + tan (x)/ *tan (x)*e   - -------------------- + x*tan (x)*\1 + tan (x)/*e   - 8*x*\1 + tan (x)/ *tan (x)*e   - 4*x*tan (x)*\1 + tan (x)/*e   - 4*\1 + tan (x)/ *tan (x)*(-1 + x)*e   - 2*tan (x)*\1 + tan (x)/*(-1 + x)*e   + 2*tan (x)*\1 + tan (x)/*(-2 + x)*e   + 4*x*tan (x)*\1 + tan (x)/*e   + 12*x*\1 + tan (x)/ *tan (x)*e   + 26*x*\1 + tan (x)/ *tan (x)*e  
                                                                                                                                                                                                                          5
12x(tan⁡2(x)+1)3e−xtan⁡2(x)+26x(tan⁡2(x)+1)2e−xtan⁡4(x)−8x(tan⁡2(x)+1)2e−xtan⁡3(x)+4x(tan⁡2(x)+1)e−xtan⁡6(x)−4x(tan⁡2(x)+1)e−xtan⁡5(x)+x(tan⁡2(x)+1)e−xtan⁡4(x)−(x−3)e−xtan⁡5(x)5+2(x−2)(tan⁡2(x)+1)e−xtan⁡4(x)−4(x−1)(tan⁡2(x)+1)2e−xtan⁡3(x)−2(x−1)(tan⁡2(x)+1)e−xtan⁡5(x)+4(tan⁡2(x)+1)3+28(tan⁡2(x)+1)2tan⁡2(x)+2(tan⁡2(x)+1)2+8(tan⁡2(x)+1)2e−xtan⁡3(x)+8(tan⁡2(x)+1)tan⁡4(x)+4(tan⁡2(x)+1)tan⁡2(x)+4(tan⁡2(x)+1)e−xtan⁡5(x)−2(tan⁡2(x)+1)e−xtan⁡4(x)12 x \left(\tan^{2}{\left(x \right)} + 1\right)^{3} e^{- x} \tan^{2}{\left(x \right)} + 26 x \left(\tan^{2}{\left(x \right)} + 1\right)^{2} e^{- x} \tan^{4}{\left(x \right)} - 8 x \left(\tan^{2}{\left(x \right)} + 1\right)^{2} e^{- x} \tan^{3}{\left(x \right)} + 4 x \left(\tan^{2}{\left(x \right)} + 1\right) e^{- x} \tan^{6}{\left(x \right)} - 4 x \left(\tan^{2}{\left(x \right)} + 1\right) e^{- x} \tan^{5}{\left(x \right)} + x \left(\tan^{2}{\left(x \right)} + 1\right) e^{- x} \tan^{4}{\left(x \right)} - \frac{\left(x - 3\right) e^{- x} \tan^{5}{\left(x \right)}}{5} + 2 \left(x - 2\right) \left(\tan^{2}{\left(x \right)} + 1\right) e^{- x} \tan^{4}{\left(x \right)} - 4 \left(x - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)^{2} e^{- x} \tan^{3}{\left(x \right)} - 2 \left(x - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) e^{- x} \tan^{5}{\left(x \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right)^{3} + 28 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan^{2}{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 8 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} e^{- x} \tan^{3}{\left(x \right)} + 8 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{4}{\left(x \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) e^{- x} \tan^{5}{\left(x \right)} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) e^{- x} \tan^{4}{\left(x \right)}
Simplificar
Gráfico
Derivada de x*exp(-x)1/5tg^5+2/3tg^3x+tgx
    © Sr Examen – Калькулятор