Mister Exam

Derivative of 2^(x/ln(x))

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   x   
 ------
 log(x)
2      
$$2^{\frac{x}{\log{\left(x \right)}}}$$
2^(x/log(x))
Detail solution
  1. Let .

  2. Then, apply the chain rule. Multiply by :

    1. Apply the quotient rule, which is:

      and .

      To find :

      1. Apply the power rule: goes to

      To find :

      1. The derivative of is .

      Now plug in to the quotient rule:

    The result of the chain rule is:


The answer is:

The graph
The first derivative [src]
   x                             
 ------                          
 log(x) /  1         1   \       
2      *|------ - -------|*log(2)
        |log(x)      2   |       
        \         log (x)/       
$$2^{\frac{x}{\log{\left(x \right)}}} \left(\frac{1}{\log{\left(x \right)}} - \frac{1}{\log{\left(x \right)}^{2}}\right) \log{\left(2 \right)}$$
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)                      
$$\frac{2^{\frac{x}{\log{\left(x \right)}}} \left(\left(1 - \frac{1}{\log{\left(x \right)}}\right)^{2} \log{\left(2 \right)} - \frac{1 - \frac{2}{\log{\left(x \right)}}}{x}\right) \log{\left(2 \right)}}{\log{\left(x \right)}^{2}}$$
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)                                         
$$\frac{2^{\frac{x}{\log{\left(x \right)}}} \left(\frac{\left(1 - \frac{1}{\log{\left(x \right)}}\right)^{3} \log{\left(2 \right)}^{2}}{\log{\left(x \right)}} - \frac{3 \left(1 - \frac{2}{\log{\left(x \right)}}\right) \left(1 - \frac{1}{\log{\left(x \right)}}\right) \log{\left(2 \right)}}{x \log{\left(x \right)}} + \frac{1 - \frac{6}{\log{\left(x \right)}^{2}}}{x^{2}}\right) \log{\left(2 \right)}}{\log{\left(x \right)}^{2}}$$
The graph
Derivative of 2^(x/ln(x))