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

1. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = x$ 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:

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

2. Now simplify:

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

The answer is:

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