Derivative of -exp(1-2x^2)

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

   1. Let $u = 1 - 2 x^{2}$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - 2 x^{2}\right)$:

      1. Differentiate $1 - 2 x^{2}$ term by term:

         1. The derivative of the constant $1$ is zero.

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

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

            So, the result is: $- 4 x$

         The result is: $- 4 x$

      The result of the chain rule is:

      $$- 4 x e^{1 - 2 x^{2}}$$

   So, the result is: $4 x e^{1 - 2 x^{2}}$

The answer is:

$$4 x e^{1 - 2 x^{2}}$$