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xe^(1/x)

Derivative of xe^(1/x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
  x ___
x*\/ e 
$$x e^{1 \cdot \frac{1}{x}}$$
d /  x ___\
--\x*\/ e /
dx         
$$\frac{d}{d x} x e^{1 \cdot \frac{1}{x}}$$
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Apply the power rule: goes to

    ; to find :

    1. Let .

    2. The derivative of is itself.

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

      1. Apply the quotient rule, which is:

        and .

        To find :

        1. The derivative of the constant is zero.

        To find :

        1. Apply the power rule: goes to

        Now plug in to the quotient rule:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
         1
         -
         x
x ___   e 
\/ e  - --
        x 
$$e^{1 \cdot \frac{1}{x}} - \frac{e^{\frac{1}{x}}}{x}$$
The second derivative [src]
 1
 -
 x
e 
--
 3
x 
$$\frac{e^{\frac{1}{x}}}{x^{3}}$$
The third derivative [src]
            1
            -
/  1    3\  x
|- -- - -|*e 
|   2   x|   
\  x     /   
-------------
       3     
      x      
$$\frac{\left(- \frac{3}{x} - \frac{1}{x^{2}}\right) e^{\frac{1}{x}}}{x^{3}}$$
The graph
Derivative of xe^(1/x)