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Identical expressions
x*e^((- one)/x^ three)
x multiply by e to the power of (( minus 1) divide by x cubed )
x multiply by e to the power of (( minus one) divide by x to the power of three)
x*e((-1)/x3)
x*e-1/x3
x*e^((-1)/x³)
x*e to the power of ((-1)/x to the power of 3)
xe^((-1)/x^3)
xe((-1)/x3)
xe-1/x3
xe^-1/x^3
x*e^((-1) divide by x^3)
Similar expressions
x*e^((1)/x^3)

The derivative of the function/ x*e^((-1)/x^3)

Derivative of x*e^((-1)/x^3)

The solution

You have entered [src]

   -1 
   ---
     3
    x 
x*E

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

x*E^(-1/x^3)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x$ and $g{\left(x \right)} = e^{\frac{1}{x^{3}}}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{1}{x^{3}}$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{1}{x^{3}}$ :
  1. Let $u = x^{3}$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{3}$ :
    1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
    The result of the chain rule is:
    
    $- \frac{3}{x^{4}}$
  The result of the chain rule is:
  
  $- \frac{3 e^{\frac{1}{x^{3}}}}{x^{4}}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The first derivative [src]

          -1 
 -1       ---
 ---        3
   3       x 
  x    3*e   
E    + ------
          3  
         x

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

Simplify

The second derivative [src]

             -1 
             ---
               3
  /     3 \   x 
3*|-2 + --|*e   
  |      3|     
  \     x /     
----------------
        4       
       x

$$\frac{3 \left(-2 + \frac{3}{x^{3}}\right) e^{- \frac{1}{x^{3}}}}{x^{4}}$$

Simplify

The third derivative [src]

                 -1 
                 ---
                   3
  /    27   9 \   x 
3*|8 - -- + --|*e   
  |     3    6|     
  \    x    x /     
--------------------
          5         
         x

$$\frac{3 \left(8 - \frac{27}{x^{3}} + \frac{9}{x^{6}}\right) e^{- \frac{1}{x^{3}}}}{x^{5}}$$

Simplify

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Derivative of x*e^((-1)/x^3)

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