Derivative of 1-e^(2x)-2xe^(2x)

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

   1. The derivative of the constant $1$ is zero.

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

      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:

         $$2 e^{2 x}$$

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

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

      1. 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 $1$

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

         So, the result is: $4 x e^{2 x} + 2 e^{2 x}$

      So, the result is: $- 4 x e^{2 x} - 2 e^{2 x}$

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

2. Now simplify:

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

The answer is: