Derivada de (x*lnx)/(lnx/x)

Solución

Ha introducido [src]

x*log(x)
--------
/log(x)\
|------|
\  x   /

$$\frac{x \log{\left(x \right)}}{\frac{1}{x} \log{\left(x \right)}}$$

(x*log(x))/((log(x)/x))

Solución detallada

Se aplica la regla de la derivada parcial:

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

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

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{2}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Según el principio, aplicamos: $x^{2}$ tenemos $2 x$
  $g{\left(x \right)} = \log{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Derivado $\log{\left(x \right)}$ es $\frac{1}{x}$ .
  Como resultado de: $2 x \log{\left(x \right)} + x$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. Derivado $\log{\left(x \right)}$ es $\frac{1}{x}$ .
Ahora aplicamos la regla de la derivada de una divesión:

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

$2 x$

Respuesta:

$2 x$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

                                                                                                       /      1   \                 /      2   \                                /             /      1   \                 -1 + log(x)\         
                                                                                                       |1 - ------|*(-1 + log(x))   |1 - ------|*(-1 + log(x))   3*(1 + log(x))*|2 - log(x) + |1 - ------|*(-1 + log(x)) - -----------|         
     /      1   \                 /      1   \                   3*(-3 + 2*log(x))   3*(-1 + log(x))   \    log(x)/                 \    log(x)/                                \             \    log(x)/                    log(x)  /         
-4 + |1 - ------|*(-1 + log(x)) - |1 - ------|*(-3 + 2*log(x)) + ----------------- + --------------- - -------------------------- - -------------------------- + ---------------------------------------------------------------------- + log(x)
     \    log(x)/                 \    log(x)/                         log(x)               2                    log(x)                       log(x)                                             log(x)                                         
                                                                                         log (x)                                                                                                                                                
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                    x*log(x)

$$\frac{- \frac{\left(1 - \frac{2}{\log{\left(x \right)}}\right) \left(\log{\left(x \right)} - 1\right)}{\log{\left(x \right)}} + \left(1 - \frac{1}{\log{\left(x \right)}}\right) \left(\log{\left(x \right)} - 1\right) - \frac{\left(1 - \frac{1}{\log{\left(x \right)}}\right) \left(\log{\left(x \right)} - 1\right)}{\log{\left(x \right)}} - \left(1 - \frac{1}{\log{\left(x \right)}}\right) \left(2 \log{\left(x \right)} - 3\right) + \frac{3 \left(\log{\left(x \right)} - 1\right)}{\log{\left(x \right)}^{2}} + \frac{3 \left(\log{\left(x \right)} + 1\right) \left(\left(1 - \frac{1}{\log{\left(x \right)}}\right) \left(\log{\left(x \right)} - 1\right) - \frac{\log{\left(x \right)} - 1}{\log{\left(x \right)}} - \log{\left(x \right)} + 2\right)}{\log{\left(x \right)}} + \frac{3 \left(2 \log{\left(x \right)} - 3\right)}{\log{\left(x \right)}} + \log{\left(x \right)} - 4}{x \log{\left(x \right)}}$$

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Gráfico

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Derivada de (x*lnx)/(lnx/x)

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