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x*exp(-x)lnsinx^7
Derivada de la función/ x*exp/ lnsinx/ exp(-x)/ x*exp(-x)lnsinx^7

Derivada de x*exp(-x)lnsinx^7

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

Gráfico:

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Solución

Ha introducido [src] 
   -x           7   
x*e  *log(x)*sin (x)
xe−xlog⁡(x)sin⁡7(x)x e^{- x} \log{\left(x \right)} \sin^{7}{\left(x \right)} 
((x*exp(-x))*log(x))*sin(x)^7
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)=xlog⁡(x)sin⁡7(x)f{\left(x \right)} = x \log{\left(x \right)} \sin^{7}{\left(x \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)h(x)=f(x)g(x)ddxh(x)+f(x)h(x)ddxg(x)+g(x)h(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} h{\left(x \right)} = f{\left(x \right)} g{\left(x \right)} \frac{d}{d x} h{\left(x \right)} + f{\left(x \right)} h{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} h{\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⁡7(x)g{\left(x \right)} = \sin^{7}{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

      2. Según el principio, aplicamos: u7u^{7} tenemos 7u67 u^{6}

      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:

        7sin⁡6(x)cos⁡(x)7 \sin^{6}{\left(x \right)} \cos{\left(x \right)}

      h(x)=log⁡(x)h{\left(x \right)} = \log{\left(x \right)}; calculamos ddxh(x)\frac{d}{d x} h{\left(x \right)}:

      1. Derivado log⁡(x)\log{\left(x \right)} es 1x\frac{1}{x}.

      Como resultado de: 7xlog⁡(x)sin⁡6(x)cos⁡(x)+log⁡(x)sin⁡7(x)+sin⁡7(x)7 x \log{\left(x \right)} \sin^{6}{\left(x \right)} \cos{\left(x \right)} + \log{\left(x \right)} \sin^{7}{\left(x \right)} + \sin^{7}{\left(x \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:

    (−xexlog⁡(x)sin⁡7(x)+(7xlog⁡(x)sin⁡6(x)cos⁡(x)+log⁡(x)sin⁡7(x)+sin⁡7(x))ex)e−2x\left(- x e^{x} \log{\left(x \right)} \sin^{7}{\left(x \right)} + \left(7 x \log{\left(x \right)} \sin^{6}{\left(x \right)} \cos{\left(x \right)} + \log{\left(x \right)} \sin^{7}{\left(x \right)} + \sin^{7}{\left(x \right)}\right) e^{x}\right) e^{- 2 x}

  2. Simplificamos:

    (−xlog⁡(x)sin⁡(x)+7xlog⁡(x)cos⁡(x)+log⁡(x)sin⁡(x)+sin⁡(x))e−xsin⁡6(x)\left(- x \log{\left(x \right)} \sin{\left(x \right)} + 7 x \log{\left(x \right)} \cos{\left(x \right)} + \log{\left(x \right)} \sin{\left(x \right)} + \sin{\left(x \right)}\right) e^{- x} \sin^{6}{\left(x \right)}

Respuesta:

(−xlog⁡(x)sin⁡(x)+7xlog⁡(x)cos⁡(x)+log⁡(x)sin⁡(x)+sin⁡(x))e−xsin⁡6(x)\left(- x \log{\left(x \right)} \sin{\left(x \right)} + 7 x \log{\left(x \right)} \cos{\left(x \right)} + \log{\left(x \right)} \sin{\left(x \right)} + \sin{\left(x \right)}\right) e^{- x} \sin^{6}{\left(x \right)}

Gráfica
02468-8-6-4-2-10100.5-0.5
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Primera derivada [src] 
   7    //     -x    -x\           -x\          6            -x       
sin (x)*\\- x*e   + e  /*log(x) + e  / + 7*x*sin (x)*cos(x)*e  *log(x)
7xe−xlog⁡(x)sin⁡6(x)cos⁡(x)+((−xe−x+e−x)log⁡(x)+e−x)sin⁡7(x)7 x e^{- x} \log{\left(x \right)} \sin^{6}{\left(x \right)} \cos{\left(x \right)} + \left(\left(- x e^{- x} + e^{- x}\right) \log{\left(x \right)} + e^{- x}\right) \sin^{7}{\left(x \right)}
Simplificar
Segunda derivada [src] 
    5    /   2    /1                     2*(-1 + x)\       /   2           2   \                                                 \  -x
-sin (x)*|sin (x)*|- - (-2 + x)*log(x) + ----------| + 7*x*\sin (x) - 6*cos (x)/*log(x) + 14*(-1 + (-1 + x)*log(x))*cos(x)*sin(x)|*e  
         \        \x                         x     /                                                                             /
−(7x(sin⁡2(x)−6cos⁡2(x))log⁡(x)+14((x−1)log⁡(x)−1)sin⁡(x)cos⁡(x)+(−(x−2)log⁡(x)+2(x−1)x+1x)sin⁡2(x))e−xsin⁡5(x)- \left(7 x \left(\sin^{2}{\left(x \right)} - 6 \cos^{2}{\left(x \right)}\right) \log{\left(x \right)} + 14 \left(\left(x - 1\right) \log{\left(x \right)} - 1\right) \sin{\left(x \right)} \cos{\left(x \right)} + \left(- \left(x - 2\right) \log{\left(x \right)} + \frac{2 \left(x - 1\right)}{x} + \frac{1}{x}\right) \sin^{2}{\left(x \right)}\right) e^{- x} \sin^{5}{\left(x \right)}
Simplificar
Tercera derivada [src] 
   4    /   3    /2                      3*(-2 + x)   3*(-1 + x)\         2    /1                     2*(-1 + x)\                                    /   2           2   \              /        2            2   \              \  -x
sin (x)*|sin (x)*|-- - (-3 + x)*log(x) + ---------- + ----------| - 21*sin (x)*|- - (-2 + x)*log(x) + ----------|*cos(x) + 21*(-1 + (-1 + x)*log(x))*\sin (x) - 6*cos (x)/*sin(x) - 7*x*\- 30*cos (x) + 19*sin (x)/*cos(x)*log(x)|*e  
        |        | 2                         x             2    |              \x                         x     /                                                                                                                |    
        \        \x                                       x     /                                                                                                                                                                /
(−7x(19sin⁡2(x)−30cos⁡2(x))log⁡(x)cos⁡(x)+21((x−1)log⁡(x)−1)(sin⁡2(x)−6cos⁡2(x))sin⁡(x)−21(−(x−2)log⁡(x)+2(x−1)x+1x)sin⁡2(x)cos⁡(x)+(−(x−3)log⁡(x)+3(x−2)x+3(x−1)x2+2x2)sin⁡3(x))e−xsin⁡4(x)\left(- 7 x \left(19 \sin^{2}{\left(x \right)} - 30 \cos^{2}{\left(x \right)}\right) \log{\left(x \right)} \cos{\left(x \right)} + 21 \left(\left(x - 1\right) \log{\left(x \right)} - 1\right) \left(\sin^{2}{\left(x \right)} - 6 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} - 21 \left(- \left(x - 2\right) \log{\left(x \right)} + \frac{2 \left(x - 1\right)}{x} + \frac{1}{x}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \left(- \left(x - 3\right) \log{\left(x \right)} + \frac{3 \left(x - 2\right)}{x} + \frac{3 \left(x - 1\right)}{x^{2}} + \frac{2}{x^{2}}\right) \sin^{3}{\left(x \right)}\right) e^{- x} \sin^{4}{\left(x \right)}
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Gráfico
Derivada de x*exp(-x)lnsinx^7
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