Derivative of 2^(x/lnx)

The solution

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

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

2^(x/log(x))

Detail solution

Let $u = \frac{x}{\log{\left(x \right)}}$ .
$\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{\log{\left(x \right)}}$ :
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = x$ and $g{\left(x \right)} = \log{\left(x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  Now plug in to the quotient rule:
  
  $\frac{\log{\left(x \right)} - 1}{\log{\left(x \right)}^{2}}$
The result of the chain rule is:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Identical expressions

Derivative of 2^(x/lnx)

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Piecewise:

The solution