Sr Examen

Derivada de y=x^lnx/e^x

Función f() - derivada -er orden en el punto
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Gráfico:

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Definida a trozos:

Solución

Ha introducido [src]
 log(x)
x      
-------
    x  
   E   
xlog(x)ex\frac{x^{\log{\left(x \right)}}}{e^{x}}
x^log(x)/E^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)=xlog(x)f{\left(x \right)} = x^{\log{\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. No logro encontrar los pasos en la búsqueda de esta derivada.

      Perola derivada

      (log(log(x))+1)log(x)log(x)\left(\log{\left(\log{\left(x \right)} \right)} + 1\right) \log{\left(x \right)}^{\log{\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:

    (xlog(x)ex+(log(log(x))+1)exlog(x)log(x))e2x\left(- x^{\log{\left(x \right)}} e^{x} + \left(\log{\left(\log{\left(x \right)} \right)} + 1\right) e^{x} \log{\left(x \right)}^{\log{\left(x \right)}}\right) e^{- 2 x}

  2. Simplificamos:

    (xlog(x)+(log(log(x))+1)log(x)log(x))ex\left(- x^{\log{\left(x \right)}} + \left(\log{\left(\log{\left(x \right)} \right)} + 1\right) \log{\left(x \right)}^{\log{\left(x \right)}}\right) e^{- x}


Respuesta:

(xlog(x)+(log(log(x))+1)log(x)log(x))ex\left(- x^{\log{\left(x \right)}} + \left(\log{\left(\log{\left(x \right)} \right)} + 1\right) \log{\left(x \right)}^{\log{\left(x \right)}}\right) e^{- x}

Gráfica
02468-8-6-4-2-1010-1000010000
Primera derivada [src]
                   log(x)  -x       
   log(x)  -x   2*x      *e  *log(x)
- x      *e   + --------------------
                         x          
xlog(x)ex+2xlog(x)exlog(x)x- x^{\log{\left(x \right)}} e^{- x} + \frac{2 x^{\log{\left(x \right)}} e^{- x} \log{\left(x \right)}}{x}
Segunda derivada [src]
        /                 /                  2   \\    
 log(x) |    4*log(x)   2*\1 - log(x) + 2*log (x)/|  -x
x      *|1 - -------- + --------------------------|*e  
        |       x                    2            |    
        \                           x             /    
xlog(x)(14log(x)x+2(2log(x)2log(x)+1)x2)exx^{\log{\left(x \right)}} \left(1 - \frac{4 \log{\left(x \right)}}{x} + \frac{2 \left(2 \log{\left(x \right)}^{2} - \log{\left(x \right)} + 1\right)}{x^{2}}\right) e^{- x}
Tercera derivada [src]
        /       /                  2   \     /          2           3              \           \    
 log(x) |     6*\1 - log(x) + 2*log (x)/   2*\-3 - 6*log (x) + 4*log (x) + 8*log(x)/   6*log(x)|  -x
x      *|-1 - -------------------------- + ----------------------------------------- + --------|*e  
        |                  2                                    3                         x    |    
        \                 x                                    x                               /    
xlog(x)(1+6log(x)x6(2log(x)2log(x)+1)x2+2(4log(x)36log(x)2+8log(x)3)x3)exx^{\log{\left(x \right)}} \left(-1 + \frac{6 \log{\left(x \right)}}{x} - \frac{6 \left(2 \log{\left(x \right)}^{2} - \log{\left(x \right)} + 1\right)}{x^{2}} + \frac{2 \left(4 \log{\left(x \right)}^{3} - 6 \log{\left(x \right)}^{2} + 8 \log{\left(x \right)} - 3\right)}{x^{3}}\right) e^{- x}
Gráfico
Derivada de y=x^lnx/e^x