Derivative of -2xe^(-x^2)

1. The derivative of a constant times a function is the constant times the derivative of the function.

   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 $g{\left(x \right)} = e^{x^{2}}$.

      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 $u = x^{2}$.

      2. The derivative of $e^{u}$ is itself.

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{2}$:

         1. Apply the power rule: $x^{2}$ goes to $2 x$

         The result of the chain rule is:

         $$2 x e^{x^{2}}$$

      Now plug in to the quotient rule:

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

   So, the result is: $- 2 \left(- 2 x^{2} e^{x^{2}} + e^{x^{2}}\right) e^{- 2 x^{2}}$

2. Now simplify:

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

The answer is: