Derivative of x*exp(-x^2/2)

1. Apply the product rule:

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

   $f{\left(x \right)} = x$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Apply the power rule: $x$ goes to $1$

   $g{\left(x \right)} = e^{\frac{\left(-1\right) x^{2}}{2}}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = \frac{\left(-1\right) x^{2}}{2}$.

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

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

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

         1. 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: $- 2 x$

         So, the result is: $- x$

      The result of the chain rule is:

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

   The result is: $- x^{2} e^{\frac{\left(-1\right) x^{2}}{2}} + e^{\frac{\left(-1\right) x^{2}}{2}}$

2. Now simplify:

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

The answer is:

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