Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of sqrt(x^2-1)
Derivative of 4/x^3
Derivative of 3x+2
Derivative of 3e^x
Integral of d{x}:
xe^(1/x)
Graphing y =:
xe^(1/x)
Identical expressions
xe^(one /x)
xe to the power of (1 divide by x)
xe to the power of (one divide by x)
xe(1/x)
xe1/x
xe^1/x
xe^(1 divide by x)
Similar expressions
xe^1/x

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

Derivative of xe^(1/x)

The solution

You have entered [src]

  x ___
x*\/ e

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

d /  x ___\
--\x*\/ e /
dx

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

Transform

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 $1$
$g{\left(x \right)} = e^{1 \cdot \frac{1}{x}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 1 \cdot \frac{1}{x}$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 1 \cdot \frac{1}{x}$ :
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = 1$ and $g{\left(x \right)} = x$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. The derivative of the constant $1$ is zero.
    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:
    
    $- \frac{1}{x^{2}}$
  The result of the chain rule is:
  
  $- \frac{e^{\frac{1}{x}}}{x^{2}}$
The result is: $e^{1 \cdot \frac{1}{x}} - \frac{e^{\frac{1}{x}}}{x}$
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

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

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 xe^(1/x)