Derivative of xexp(x^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 $1$

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

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

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

2. Now simplify:

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

The answer is: