Derivative of (2ax+b)*e^(3*x)+2(ax^2+bx)e^(3*x)

1. Differentiate $e^{3 x} 2 \left(a x^{2} + b x\right) + e^{3 x} \left(2 a x + b\right)$ term by term:

   1. Apply the product rule:

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

      $f{\left(x \right)} = 2 a x + b$; to find $\frac{d}{d x} f{\left(x \right)}$:

      1. Differentiate $2 a x + b$ term by term:

         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: $2 a$

         2. The derivative of the constant $b$ is zero.

         The result is: $2 a$

      $g{\left(x \right)} = e^{3 x}$; 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: $2 a e^{3 x} + 3 \left(2 a x + b\right) e^{3 x}$

   2. Apply the product rule:

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

      $f{\left(x \right)} = 2 \left(a x^{2} + b x\right)$; 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. Differentiate $a x^{2} + b x$ term by term:

            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: $2 a x$

            2. 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: $b$

            The result is: $2 a x + b$

         So, the result is: $4 a x + 2 b$

      $g{\left(x \right)} = e^{3 x}$; 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: $\left(4 a x + 2 b\right) e^{3 x} + 6 \left(a x^{2} + b x\right) e^{3 x}$

   The result is: $2 a e^{3 x} + \left(4 a x + 2 b\right) e^{3 x} + 6 \left(a x^{2} + b x\right) e^{3 x} + 3 \left(2 a x + b\right) e^{3 x}$

2. Now simplify:

   $$\left(10 a x + 2 a + 5 b + 6 x \left(a x + b\right)\right) e^{3 x}$$

The answer is:

$$\left(10 a x + 2 a + 5 b + 6 x \left(a x + b\right)\right) e^{3 x}$$