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Derivada de la función/ x/lnx/ а^x/lnx

Derivada de а^x/lnx

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

Ha introducido [src] 
   x  
  a   
------
log(x)
axlog⁡(x)\frac{a^{x}}{\log{\left(x \right)}} 
a^x/log(x)
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)=axf{\left(x \right)} = a^{x} y g(x)=log⁡(x)g{\left(x \right)} = \log{\left(x \right)}.

    Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. ∂∂xax=axlog⁡(a)\frac{\partial}{\partial x} a^{x} = a^{x} \log{\left(a \right)}

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

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

    Ahora aplicamos la regla de la derivada de una divesión:

    axlog⁡(a)log⁡(x)−axxlog⁡(x)2\frac{a^{x} \log{\left(a \right)} \log{\left(x \right)} - \frac{a^{x}}{x}}{\log{\left(x \right)}^{2}}

  2. Simplificamos:

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

Respuesta:

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

Primera derivada [src] 
 x               x   
a *log(a)       a    
--------- - ---------
  log(x)         2   
            x*log (x)
axlog⁡(a)log⁡(x)−axxlog⁡(x)2\frac{a^{x} \log{\left(a \right)}}{\log{\left(x \right)}} - \frac{a^{x}}{x \log{\left(x \right)}^{2}}
Simplificar
Segunda derivada [src] 
   /                2              \
   |          1 + ------           |
 x |   2          log(x)   2*log(a)|
a *|log (a) + ---------- - --------|
   |           2           x*log(x)|
   \          x *log(x)            /
------------------------------------
               log(x)
ax(log⁡(a)2−2log⁡(a)xlog⁡(x)+1+2log⁡(x)x2log⁡(x))log⁡(x)\frac{a^{x} \left(\log{\left(a \right)}^{2} - \frac{2 \log{\left(a \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(a)|
 x |   3      3*log (a)     \             log (x)/     \    log(x)/       |
a *|log (a) - --------- - ------------------------ + ---------------------|
   |           x*log(x)           3                         2             |
   \                             x *log(x)                 x *log(x)      /
---------------------------------------------------------------------------
                                   log(x)
ax(log⁡(a)3−3log⁡(a)2xlog⁡(x)+3(1+2log⁡(x))log⁡(a)x2log⁡(x)−2(1+3log⁡(x)+3log⁡(x)2)x3log⁡(x))log⁡(x)\frac{a^{x} \left(\log{\left(a \right)}^{3} - \frac{3 \log{\left(a \right)}^{2}}{x \log{\left(x \right)}} + \frac{3 \left(1 + \frac{2}{\log{\left(x \right)}}\right) \log{\left(a \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
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