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Identical expressions
x^(one /x)/x
x to the power of (1 divide by x) divide by x
x to the power of (one divide by x) divide by x
x(1/x)/x
x1/x/x
x^1/x/x
x^(1 divide by x) divide by x

The derivative of the function/ x^(1/x)/x

Derivative of x^(1/x)/x

The solution

You have entered [src]

x ___
\/ x 
-----
  x

\frac{x^{\frac{1}{x}}}{x}

x^(1/x)/x

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{\frac{1}{x}}$ and $g{\left(x \right)} = x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Don't know the steps in finding this derivative.
  
  But the derivative is
  
  $\left(\log{\left(\frac{1}{x} \right)} + 1\right) \left(\frac{1}{x}\right)^{\frac{1}{x}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
Now plug in to the quotient rule:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

       /                                  3                                                                                         \ 
       |                     (-1 + log(x))    3*(-1 + log(x))*(-3 + 2*log(x))     /                             2\                  | 
       |    -11 + 6*log(x) + -------------- + -------------------------------     |                (-1 + log(x)) |                  | 
       |                            2                        x                  3*|-3 + 2*log(x) + --------------|                  | 
 x ___ |                           x                                              \                      x       /   6*(-1 + log(x))| 
-\/ x *|6 + ----------------------------------------------------------------- + ---------------------------------- + ---------------| 
       \                                    x                                                   x                           x       / 
--------------------------------------------------------------------------------------------------------------------------------------
                                                                   4                                                                  
                                                                  x

- \frac{x^{\frac{1}{x}} \left(6 + \frac{6 \left(\log{\left(x \right)} - 1\right)}{x} + \frac{3 \left(2 \log{\left(x \right)} - 3 + \frac{\left(\log{\left(x \right)} - 1\right)^{2}}{x}\right)}{x} + \frac{6 \log{\left(x \right)} - 11 + \frac{3 \left(\log{\left(x \right)} - 1\right) \left(2 \log{\left(x \right)} - 3\right)}{x} + \frac{\left(\log{\left(x \right)} - 1\right)^{3}}{x^{2}}}{x}\right)}{x^{4}}

Simplify

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Derivative of x^(1/x)/x

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The solution