Derivada de y=2^x/lnx

Solución

Ha introducido [src]

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

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

2^x/log(x)

Solución detallada

Se aplica la regla de la derivada parcial:

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

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

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. $\frac{d}{d x} 2^{x} = 2^{x} \log{\left(2 \right)}$
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:

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

Gráfico

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Derivada de y=2^x/lnx

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