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 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: $2 x^{2} e^{x^{2}} + e^{x^{2}}$

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

2. Now simplify:

   $$\left(4 x^{2} + 2\right) e^{x^{2}}$$

The answer is: