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Derivada de:
Derivada de x^-0,2
Derivada de e-x
Derivada de e^e
Derivada de e^((3*x)^2)
Expresiones idénticas
(x^ln2x)/(ln2x)^x
(x en el grado ln2x) dividir por (ln2x) en el grado x
(xln2x)/(ln2x)x
xln2x/ln2xx
x^ln2x/ln2x^x
(x^ln2x) dividir por (ln2x)^x

Derivada de la función/ x^ln2/ (x^ln2x)/(ln2x)^x

Derivada de (x^ln2x)/(ln2x)^x

Solución

Ha introducido [src]

 log(2*x)
x        
---------
   x     
log (2*x)

$$\frac{x^{\log{\left(2 x \right)}}}{\log{\left(2 x \right)}^{x}}$$

x^log(2*x)/log(2*x)^x

Solución detallada

Se aplica la regla de la derivada parcial:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x^{\log{\left(2 x \right)}}$ y $g{\left(x \right)} = \log{\left(2 x \right)}^{x}$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $\left(\log{\left(\log{\left(2 x \right)} \right)} + 1\right) \log{\left(2 x \right)}^{\log{\left(2 x \right)}}$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $x^{x} \left(\log{\left(x \right)} + 1\right)$
Ahora aplicamos la regla de la derivada de una divesión:

$\left(- x^{x} x^{\log{\left(2 x \right)}} \left(\log{\left(x \right)} + 1\right) + \left(\log{\left(\log{\left(2 x \right)} \right)} + 1\right) \log{\left(2 x \right)}^{x} \log{\left(2 x \right)}^{\log{\left(2 x \right)}}\right) \log{\left(2 x \right)}^{- 2 x}$
Simplificamos:

$\left(- x^{x + \log{\left(2 x \right)}} \left(\log{\left(x \right)} + 1\right) + \left(\log{\left(\log{\left(2 x \right)} \right)} + 1\right) \log{\left(2 x \right)}^{x + \log{\left(2 x \right)}}\right) \log{\left(2 x \right)}^{- 2 x}$

Respuesta:

$\left(- x^{x + \log{\left(2 x \right)}} \left(\log{\left(x \right)} + 1\right) + \left(\log{\left(\log{\left(2 x \right)} \right)} + 1\right) \log{\left(2 x \right)}^{x + \log{\left(2 x \right)}}\right) \log{\left(2 x \right)}^{- 2 x}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

 log(2*x)    -x      /     1                    \    log(2*x)    -x      /log(x)   log(2*x)\
x        *log  (2*x)*|- -------- - log(log(2*x))| + x        *log  (2*x)*|------ + --------|
                     \  log(2*x)                /                        \  x         x    /

$$x^{\log{\left(2 x \right)}} \left(\frac{\log{\left(x \right)}}{x} + \frac{\log{\left(2 x \right)}}{x}\right) \log{\left(2 x \right)}^{- x} + x^{\log{\left(2 x \right)}} \left(- \log{\left(\log{\left(2 x \right)} \right)} - \frac{1}{\log{\left(2 x \right)}}\right) \log{\left(2 x \right)}^{- x}$$

Simplificar

Segunda derivada [src]

                     /                                                                                    1         /   1                    \                    \
                     |                          2                          2                       1 - --------   2*|-------- + log(log(2*x))|*(log(x) + log(2*x))|
 log(2*x)    -x      |/   1                    \    2 + (log(x) + log(2*x))  - log(x) - log(2*x)       log(2*x)     \log(2*x)                /                    |
x        *log  (2*x)*||-------- + log(log(2*x))|  + -------------------------------------------- - ------------ - ------------------------------------------------|
                     |\log(2*x)                /                          2                         x*log(2*x)                           x                        |
                     \                                                   x                                                                                        /

$$x^{\log{\left(2 x \right)}} \left(\left(\log{\left(\log{\left(2 x \right)} \right)} + \frac{1}{\log{\left(2 x \right)}}\right)^{2} - \frac{1 - \frac{1}{\log{\left(2 x \right)}}}{x \log{\left(2 x \right)}} - \frac{2 \left(\log{\left(x \right)} + \log{\left(2 x \right)}\right) \left(\log{\left(\log{\left(2 x \right)} \right)} + \frac{1}{\log{\left(2 x \right)}}\right)}{x} + \frac{\left(\log{\left(x \right)} + \log{\left(2 x \right)}\right)^{2} - \log{\left(x \right)} - \log{\left(2 x \right)} + 2}{x^{2}}\right) \log{\left(2 x \right)}^{- x}$$

Simplificar

Tercera derivada [src]

                     /                                                                                                                                                                                                                                     /                                     1    \                                                                  \
                     |                                                                                                                                             2                                                                                       |                          2   1 - --------|                                                                  |
                     |                                                                                                                                     1 - ---------     /   1                    \ /                       2                    \     |/   1                    \        log(2*x)|                         /       1    \ /   1                    \|
                     |                            3                           3                                                                                   2        3*|-------- + log(log(2*x))|*\2 + (log(x) + log(2*x))  - log(x) - log(2*x)/   3*||-------- + log(log(2*x))|  - ------------|*(log(x) + log(2*x))   3*|1 - --------|*|-------- + log(log(2*x))||
 log(2*x)    -x      |  /   1                    \    -6 + (log(x) + log(2*x))  + 2*log(x) + 2*log(2*x) - 3*(log(x) + log(2*x))*(-2 + log(x) + log(2*x))       log (2*x)     \log(2*x)                /                                                    \\log(2*x)                /     x*log(2*x) /                         \    log(2*x)/ \log(2*x)                /|
x        *log  (2*x)*|- |-------- + log(log(2*x))|  + -------------------------------------------------------------------------------------------------- + ------------- - --------------------------------------------------------------------------- + ------------------------------------------------------------------ + -------------------------------------------|
                     |  \log(2*x)                /                                                     3                                                     2                                                   2                                                                       x                                                     x*log(2*x)                |
                     \                                                                                x                                                     x *log(2*x)                                         x                                                                                                                                                        /

$$x^{\log{\left(2 x \right)}} \left(- \left(\log{\left(\log{\left(2 x \right)} \right)} + \frac{1}{\log{\left(2 x \right)}}\right)^{3} + \frac{3 \left(1 - \frac{1}{\log{\left(2 x \right)}}\right) \left(\log{\left(\log{\left(2 x \right)} \right)} + \frac{1}{\log{\left(2 x \right)}}\right)}{x \log{\left(2 x \right)}} + \frac{3 \left(\left(\log{\left(\log{\left(2 x \right)} \right)} + \frac{1}{\log{\left(2 x \right)}}\right)^{2} - \frac{1 - \frac{1}{\log{\left(2 x \right)}}}{x \log{\left(2 x \right)}}\right) \left(\log{\left(x \right)} + \log{\left(2 x \right)}\right)}{x} + \frac{1 - \frac{2}{\log{\left(2 x \right)}^{2}}}{x^{2} \log{\left(2 x \right)}} - \frac{3 \left(\log{\left(\log{\left(2 x \right)} \right)} + \frac{1}{\log{\left(2 x \right)}}\right) \left(\left(\log{\left(x \right)} + \log{\left(2 x \right)}\right)^{2} - \log{\left(x \right)} - \log{\left(2 x \right)} + 2\right)}{x^{2}} + \frac{\left(\log{\left(x \right)} + \log{\left(2 x \right)}\right)^{3} - 3 \left(\log{\left(x \right)} + \log{\left(2 x \right)}\right) \left(\log{\left(x \right)} + \log{\left(2 x \right)} - 2\right) + 2 \log{\left(x \right)} + 2 \log{\left(2 x \right)} - 6}{x^{3}}\right) \log{\left(2 x \right)}^{- x}$$

Simplificar

Gráfico

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Expresiones idénticas

Derivada de (x^ln2x)/(ln2x)^x

Gráfico:

Definida a trozos:

Solución