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((x^x)+ln(x))^(1/4)

Derivada de ((x^x)+ln(x))^(1/4)

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

Ha introducido [src]
   _____________
4 /  x          
\/  x  + log(x) 
xx+log(x)4\sqrt[4]{x^{x} + \log{\left(x \right)}}
(x^x + log(x))^(1/4)
Solución detallada
  1. Sustituimos u=xx+log(x)u = x^{x} + \log{\left(x \right)}.

  2. Según el principio, aplicamos: u4\sqrt[4]{u} tenemos 14u34\frac{1}{4 u^{\frac{3}{4}}}

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddx(xx+log(x))\frac{d}{d x} \left(x^{x} + \log{\left(x \right)}\right):

    1. diferenciamos xx+log(x)x^{x} + \log{\left(x \right)} miembro por miembro:

      1. No logro encontrar los pasos en la búsqueda de esta derivada.

        Perola derivada

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

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

      Como resultado de: xx(log(x)+1)+1xx^{x} \left(\log{\left(x \right)} + 1\right) + \frac{1}{x}

    Como resultado de la secuencia de reglas:

    xx(log(x)+1)+1x4(xx+log(x))34\frac{x^{x} \left(\log{\left(x \right)} + 1\right) + \frac{1}{x}}{4 \left(x^{x} + \log{\left(x \right)}\right)^{\frac{3}{4}}}

  4. Simplificamos:

    xx+1(log(x)+1)+14x(xx+log(x))34\frac{x^{x + 1} \left(\log{\left(x \right)} + 1\right) + 1}{4 x \left(x^{x} + \log{\left(x \right)}\right)^{\frac{3}{4}}}


Respuesta:

xx+1(log(x)+1)+14x(xx+log(x))34\frac{x^{x + 1} \left(\log{\left(x \right)} + 1\right) + 1}{4 x \left(x^{x} + \log{\left(x \right)}\right)^{\frac{3}{4}}}

Gráfica
02468-8-6-4-2-10100500
Primera derivada [src]
       x             
 1    x *(1 + log(x))
--- + ---------------
4*x          4       
---------------------
                3/4  
   / x         \     
   \x  + log(x)/     
xx(log(x)+1)4+14x(xx+log(x))34\frac{\frac{x^{x} \left(\log{\left(x \right)} + 1\right)}{4} + \frac{1}{4 x}}{\left(x^{x} + \log{\left(x \right)}\right)^{\frac{3}{4}}}
Segunda derivada [src]
                              2                            
         /1    x             \                             
       3*|- + x *(1 + log(x))|       x                     
  4      \x                  /    4*x       x             2
- -- - ------------------------ + ---- + 4*x *(1 + log(x)) 
   2          x                    x                       
  x          x  + log(x)                                   
-----------------------------------------------------------
                                    3/4                    
                       / x         \                       
                    16*\x  + log(x)/                       
4xx(log(x)+1)23(xx(log(x)+1)+1x)2xx+log(x)+4xxx4x216(xx+log(x))34\frac{4 x^{x} \left(\log{\left(x \right)} + 1\right)^{2} - \frac{3 \left(x^{x} \left(\log{\left(x \right)} + 1\right) + \frac{1}{x}\right)^{2}}{x^{x} + \log{\left(x \right)}} + \frac{4 x^{x}}{x} - \frac{4}{x^{2}}}{16 \left(x^{x} + \log{\left(x \right)}\right)^{\frac{3}{4}}}
Tercera derivada [src]
                                                                                        /        x                   \                     
                                                           3      /1    x             \ |  1    x     x             2|                     
                                      /1    x             \    36*|- + x *(1 + log(x))|*|- -- + -- + x *(1 + log(x)) |                     
         x                         21*|- + x *(1 + log(x))|       \x                  / |   2   x                    |       x             
32   16*x        x             3      \x                  /                             \  x                         /   48*x *(1 + log(x))
-- - ----- + 16*x *(1 + log(x))  + ------------------------- - ------------------------------------------------------- + ------------------
 3      2                                             2                               x                                          x         
x      x                                 / x         \                               x  + log(x)                                           
                                         \x  + log(x)/                                                                                     
-------------------------------------------------------------------------------------------------------------------------------------------
                                                                            3/4                                                            
                                                               / x         \                                                               
                                                            64*\x  + log(x)/                                                               
16xx(log(x)+1)336(xx(log(x)+1)+1x)(xx(log(x)+1)2+xxx1x2)xx+log(x)+21(xx(log(x)+1)+1x)3(xx+log(x))2+48xx(log(x)+1)x16xxx2+32x364(xx+log(x))34\frac{16 x^{x} \left(\log{\left(x \right)} + 1\right)^{3} - \frac{36 \left(x^{x} \left(\log{\left(x \right)} + 1\right) + \frac{1}{x}\right) \left(x^{x} \left(\log{\left(x \right)} + 1\right)^{2} + \frac{x^{x}}{x} - \frac{1}{x^{2}}\right)}{x^{x} + \log{\left(x \right)}} + \frac{21 \left(x^{x} \left(\log{\left(x \right)} + 1\right) + \frac{1}{x}\right)^{3}}{\left(x^{x} + \log{\left(x \right)}\right)^{2}} + \frac{48 x^{x} \left(\log{\left(x \right)} + 1\right)}{x} - \frac{16 x^{x}}{x^{2}} + \frac{32}{x^{3}}}{64 \left(x^{x} + \log{\left(x \right)}\right)^{\frac{3}{4}}}
Gráfico
Derivada de ((x^x)+ln(x))^(1/4)