Derivative of cx^2exp(2x)+cxexp(2x)

1. Differentiate $c x e^{2 x} + c x^{2} e^{2 x}$ term by term:

   1. Apply the product rule:

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

      $f{\left(x \right)} = c x^{2}$; 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: $2 c x$

      $g{\left(x \right)} = e^{2 x}$; to find $\frac{d}{d x} g{\left(x \right)}$:

      1. Let $u = 2 x$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$:

         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: $2$

         The result of the chain rule is:

         $$2 e^{2 x}$$

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

   2. Apply the product rule:

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

      $f{\left(x \right)} = c x$; 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: $c$

      $g{\left(x \right)} = e^{2 x}$; to find $\frac{d}{d x} g{\left(x \right)}$:

      1. Let $u = 2 x$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$:

         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: $2$

         The result of the chain rule is:

         $$2 e^{2 x}$$

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

   The result is: $2 c x^{2} e^{2 x} + 4 c x e^{2 x} + c e^{2 x}$

2. Now simplify:

   $$c \left(2 x^{2} + 4 x + 1\right) e^{2 x}$$

The answer is:

$$c \left(2 x^{2} + 4 x + 1\right) e^{2 x}$$