Derivative of x^2e^3x

1. Apply the product rule:

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

   $f{\left(x \right)} = e^{3} x^{2}$; 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: $x^{2}$ goes to $2 x$

      So, the result is: $2 x e^{3}$

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

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

   The result is: $e^{3} x^{2} + 2 x^{2} e^{3}$

2. Now simplify:

   $$3 x^{2} e^{3}$$

The answer is:

$$3 x^{2} e^{3}$$