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

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

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

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The solution

You have entered [src] 
   -1 
   ---
     3
    x 
x*E
e−1x3xe^{- \frac{1}{x^{3}}} x 
x*E^(-1/x^3)
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=xf{\left(x \right)} = x and g(x)=e1x3g{\left(x \right)} = e^{\frac{1}{x^{3}}}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the power rule: xx goes to 11

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=1x3u = \frac{1}{x^{3}}.

    2. The derivative of eue^{u} is itself.

    3. Then, apply the chain rule. Multiply by ddx1x3\frac{d}{d x} \frac{1}{x^{3}}:

      1. Let u=x3u = x^{3}.

      2. Apply the power rule: 1u\frac{1}{u} goes to −1u2- \frac{1}{u^{2}}

      3. Then, apply the chain rule. Multiply by ddxx3\frac{d}{d x} x^{3}:

        1. Apply the power rule: x3x^{3} goes to 3x23 x^{2}

        The result of the chain rule is:

        −3x4- \frac{3}{x^{4}}

      The result of the chain rule is:

      −3e1x3x4- \frac{3 e^{\frac{1}{x^{3}}}}{x^{4}}

    Now plug in to the quotient rule:

    (e1x3+3e1x3x3)e−2x3\left(e^{\frac{1}{x^{3}}} + \frac{3 e^{\frac{1}{x^{3}}}}{x^{3}}\right) e^{- \frac{2}{x^{3}}}

  2. Now simplify:

    (x3+3)e−1x3x3\frac{\left(x^{3} + 3\right) e^{- \frac{1}{x^{3}}}}{x^{3}}

The answer is:

(x3+3)e−1x3x3\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−1x3+3e−1x3x3e^{- \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
3(−2+3x3)e−1x3x4\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
3(8−27x3+9x6)e−1x3x5\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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