Derivative of 6e^(3x)+3xe^(3x)+(-x^2+x+3)*e^(3x)

1. Differentiate $\left(e^{3 x} 3 x + 6 e^{3 x}\right) + e^{3 x} \left(\left(- x^{2} + x\right) + 3\right)$ term by term:

   1. Differentiate $e^{3 x} 3 x + 6 e^{3 x}$ term by term:

      1. The derivative of a constant times a function is the constant times the derivative of the function.

         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:

            $$3 e^{3 x}$$

         So, the result is: $18 e^{3 x}$

      2. Apply the product rule:

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

         $f{\left(x \right)} = 3 x$; 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: $x$ goes to $1$

            So, the result is: $3$

         $g{\left(x \right)} = e^{3 x}$; 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: $9 x e^{3 x} + 3 e^{3 x}$

      The result is: $9 x e^{3 x} + 21 e^{3 x}$

   2. Apply the product rule:

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

      $f{\left(x \right)} = \left(- x^{2} + x\right) + 3$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

         1. Differentiate $- x^{2} + x$ 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: $- 2 x$

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

            The result is: $1 - 2 x$

         2. The derivative of the constant $3$ is zero.

         The result is: $1 - 2 x$

      $g{\left(x \right)} = e^{3 x}$; 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: $\left(1 - 2 x\right) e^{3 x} + 3 \left(\left(- x^{2} + x\right) + 3\right) e^{3 x}$

   The result is: $9 x e^{3 x} + \left(1 - 2 x\right) e^{3 x} + 3 \left(\left(- x^{2} + x\right) + 3\right) e^{3 x} + 21 e^{3 x}$

2. Now simplify:

   $$\left(- 3 x^{2} + 10 x + 31\right) e^{3 x}$$

The answer is:

$$\left(- 3 x^{2} + 10 x + 31\right) e^{3 x}$$