Derivative of ax*e^(-ax^2)

1. Apply the product rule:

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

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

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

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

      So, the result is: $a$

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

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

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

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

      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: $- 2 a x$

      The result of the chain rule is:

      $$- 2 a x e^{- a x^{2}}$$

   The result is: $- 2 a^{2} x^{2} e^{- a x^{2}} + a e^{- a x^{2}}$

2. Now simplify:

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

The answer is: