Derivative of e^(3x)+3xe^(3x)

1. Differentiate $3 x e^{3 x} + e^{3 x}$ term by term:

   1. Let $u = 3 x$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$:

      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: $3$

      The result of the chain rule is:

      $$3 e^{3 x}$$

   4. 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 $1$

         $g{\left(x \right)} = e^{3 x}$; to find $\frac{d}{d x} g{\left(x \right)}$:

         1. Let $u = 3 x$.

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$:

            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: $3$

            The result of the chain rule is:

            $$3 e^{3 x}$$

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

      So, the result is: $9 x e^{3 x} + 3 e^{3 x}$

   The result is: $9 x e^{3 x} + 6 e^{3 x}$

2. Now simplify:

   $$\left(9 x + 6\right) e^{3 x}$$

The answer is:

$$\left(9 x + 6\right) e^{3 x}$$