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(1/x)^(1/x)

Derivative of (1/x)^(1/x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
    _____
   /   1 
x /  1*- 
\/     x 
$$\left(1 \cdot \frac{1}{x}\right)^{1 \cdot \frac{1}{x}}$$
  /    _____\
d |   /   1 |
--|x /  1*- |
dx\\/     x /
$$\frac{d}{d x} \left(1 \cdot \frac{1}{x}\right)^{1 \cdot \frac{1}{x}}$$
Detail solution
  1. Don't know the steps in finding this derivative.

    But the derivative is

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
          /          /  1\\
    _____ |       log|1*-||
   /   1  |  1       \  x/|
x /  1*- *|- -- - --------|
\/     x  |   2       2   |
          \  x       x    /
$$\left(1 \cdot \frac{1}{x}\right)^{\frac{1}{x}} \left(- \frac{\log{\left(1 \cdot \frac{1}{x} \right)}}{x^{2}} - \frac{1}{x^{2}}\right)$$
The second derivative [src]
        /                           2\
        |               /       /1\\ |
    ___ |               |1 + log|-|| |
   / 1  |         /1\   \       \x// |
x /  - *|3 + 2*log|-| + -------------|
\/   x  \         \x/         x      /
--------------------------------------
                   3                  
                  x                   
$$\frac{\left(2 \log{\left(\frac{1}{x} \right)} + \frac{\left(\log{\left(\frac{1}{x} \right)} + 1\right)^{2}}{x} + 3\right) \left(\frac{1}{x}\right)^{\frac{1}{x}}}{x^{3}}$$
The third derivative [src]
         /                            3                                \ 
         |                /       /1\\      /       /1\\ /         /1\\| 
     ___ |                |1 + log|-||    3*|1 + log|-||*|3 + 2*log|-||| 
    / 1  |          /1\   \       \x//      \       \x// \         \x//| 
-x /  - *|11 + 6*log|-| + ------------- + -----------------------------| 
 \/   x  |          \x/          2                      x              | 
         \                      x                                      / 
-------------------------------------------------------------------------
                                     4                                   
                                    x                                    
$$- \frac{\left(6 \log{\left(\frac{1}{x} \right)} + \frac{3 \left(\log{\left(\frac{1}{x} \right)} + 1\right) \left(2 \log{\left(\frac{1}{x} \right)} + 3\right)}{x} + \frac{\left(\log{\left(\frac{1}{x} \right)} + 1\right)^{3}}{x^{2}} + 11\right) \left(\frac{1}{x}\right)^{\frac{1}{x}}}{x^{4}}$$
The graph
Derivative of (1/x)^(1/x)