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Derivative of:
Derivative of x^7*e^x
Derivative of 2/sqrt(x)
Derivative of x*e^(1/x)
Derivative of x*e^x^2
Limit of the function:
x*e^(1/x)
Identical expressions
x*e^(one /x)
x multiply by e to the power of (1 divide by x)
x multiply by e to the power of (one divide by x)
x*e(1/x)
x*e1/x
xe^(1/x)
xe(1/x)
xe1/x
xe^1/x
x*e^(1 divide by x)
Similar expressions
xe^1/x
xe^(1/x)

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

Derivative of x*e^(1/x)

The solution

You have entered [src]

  x ___
x*\/ E

$$e^{\frac{1}{x}} x$$

x*E^(1/x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
$g{\left(x \right)} = e^{\frac{1}{x}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{1}{x}$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{1}{x}$ :
  1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
  The result of the chain rule is:
  
  $- \frac{e^{\frac{1}{x}}}{x^{2}}$
The result is: $e^{\frac{1}{x}} - \frac{e^{\frac{1}{x}}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         1
         -
         x
x ___   e 
\/ E  - --
        x

$$e^{\frac{1}{x}} - \frac{e^{\frac{1}{x}}}{x}$$

Simplify

The second derivative [src]

 1
 -
 x
e 
--
 3
x

$$\frac{e^{\frac{1}{x}}}{x^{3}}$$

Simplify

The third derivative [src]

            1
            -
/  1    3\  x
|- -- - -|*e 
|   2   x|   
\  x     /   
-------------
       3     
      x

$$\frac{\left(- \frac{3}{x} - \frac{1}{x^{2}}\right) e^{\frac{1}{x}}}{x^{3}}$$

Simplify

The graph

Derivative of x*e^(1/x)