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x(log2^x)/(x-1)
Derivada de la función/ (x-1)/ x(log2^x)/(x-1)

Derivada de x(log2^x)/(x-1)

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

Ha introducido [src] 
     x   
x*log (2)
---------
  x - 1
xlog⁡(2)xx−1\frac{x \log{\left(2 \right)}^{x}}{x - 1} 
(x*log(2)^x)/(x - 1)
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⁡(2)xf{\left(x \right)} = x \log{\left(2 \right)}^{x} y g(x)=x−1g{\left(x \right)} = x - 1.

    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)=log⁡(2)xg{\left(x \right)} = \log{\left(2 \right)}^{x}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. ddxlog⁡(2)x=log⁡(2)xlog⁡(log⁡(2))\frac{d}{d x} \log{\left(2 \right)}^{x} = \log{\left(2 \right)}^{x} \log{\left(\log{\left(2 \right)} \right)}

      Como resultado de: xlog⁡(2)xlog⁡(log⁡(2))+log⁡(2)xx \log{\left(2 \right)}^{x} \log{\left(\log{\left(2 \right)} \right)} + \log{\left(2 \right)}^{x}

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

    1. diferenciamos x−1x - 1 miembro por miembro:

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

      2. Según el principio, aplicamos: xx tenemos 11

      Como resultado de: 11

    Ahora aplicamos la regla de la derivada de una divesión:

    −xlog⁡(2)x+(x−1)(xlog⁡(2)xlog⁡(log⁡(2))+log⁡(2)x)(x−1)2\frac{- x \log{\left(2 \right)}^{x} + \left(x - 1\right) \left(x \log{\left(2 \right)}^{x} \log{\left(\log{\left(2 \right)} \right)} + \log{\left(2 \right)}^{x}\right)}{\left(x - 1\right)^{2}}

  2. Simplificamos:

    (−x+(x−1)(xlog⁡(log⁡(2))+1))log⁡(2)x(x−1)2\frac{\left(- x + \left(x - 1\right) \left(x \log{\left(\log{\left(2 \right)} \right)} + 1\right)\right) \log{\left(2 \right)}^{x}}{\left(x - 1\right)^{2}}

Respuesta:

(−x+(x−1)(xlog⁡(log⁡(2))+1))log⁡(2)x(x−1)2\frac{\left(- x + \left(x - 1\right) \left(x \log{\left(\log{\left(2 \right)} \right)} + 1\right)\right) \log{\left(2 \right)}^{x}}{\left(x - 1\right)^{2}}

Gráfica
02468-8-6-4-2-1010-100100
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
   x           x                       x   
log (2) + x*log (2)*log(log(2))   x*log (2)
------------------------------- - ---------
             x - 1                        2
                                   (x - 1)
−xlog⁡(2)x(x−1)2+xlog⁡(2)xlog⁡(log⁡(2))+log⁡(2)xx−1- \frac{x \log{\left(2 \right)}^{x}}{\left(x - 1\right)^{2}} + \frac{x \log{\left(2 \right)}^{x} \log{\left(\log{\left(2 \right)} \right)} + \log{\left(2 \right)}^{x}}{x - 1}
Simplificar
Segunda derivada [src] 
   x    /                                  2*(1 + x*log(log(2)))      2*x   \
log (2)*|(2 + x*log(log(2)))*log(log(2)) - --------------------- + ---------|
        |                                          -1 + x                  2|
        \                                                          (-1 + x) /
-----------------------------------------------------------------------------
                                    -1 + x
(2x(x−1)2+(xlog⁡(log⁡(2))+2)log⁡(log⁡(2))−2(xlog⁡(log⁡(2))+1)x−1)log⁡(2)xx−1\frac{\left(\frac{2 x}{\left(x - 1\right)^{2}} + \left(x \log{\left(\log{\left(2 \right)} \right)} + 2\right) \log{\left(\log{\left(2 \right)} \right)} - \frac{2 \left(x \log{\left(\log{\left(2 \right)} \right)} + 1\right)}{x - 1}\right) \log{\left(2 \right)}^{x}}{x - 1}
Simplificar
Tercera derivada [src] 
   x    /   2                                  6*x      6*(1 + x*log(log(2)))   3*(2 + x*log(log(2)))*log(log(2))\
log (2)*|log (log(2))*(3 + x*log(log(2))) - --------- + --------------------- - ---------------------------------|
        |                                           3                 2                       -1 + x             |
        \                                   (-1 + x)          (-1 + x)                                           /
------------------------------------------------------------------------------------------------------------------
                                                      -1 + x
(−6x(x−1)3+(xlog⁡(log⁡(2))+3)log⁡(log⁡(2))2−3(xlog⁡(log⁡(2))+2)log⁡(log⁡(2))x−1+6(xlog⁡(log⁡(2))+1)(x−1)2)log⁡(2)xx−1\frac{\left(- \frac{6 x}{\left(x - 1\right)^{3}} + \left(x \log{\left(\log{\left(2 \right)} \right)} + 3\right) \log{\left(\log{\left(2 \right)} \right)}^{2} - \frac{3 \left(x \log{\left(\log{\left(2 \right)} \right)} + 2\right) \log{\left(\log{\left(2 \right)} \right)}}{x - 1} + \frac{6 \left(x \log{\left(\log{\left(2 \right)} \right)} + 1\right)}{\left(x - 1\right)^{2}}\right) \log{\left(2 \right)}^{x}}{x - 1}
Simplificar
Gráfico
Derivada de x(log2^x)/(x-1)
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