Derivative of 5e^(2x)-10xe^(-2x)

1. Differentiate $- e^{- 2 x} 10 x + 5 e^{2 x}$ term by term:

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

      1. Let $u = 2 x$.

      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: $10 e^{2 x}$

   2. 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 quotient rule, which is:

            $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

            $f{\left(x \right)} = x$ and $g{\left(x \right)} = e^{2 x}$.

            To find $\frac{d}{d x} f{\left(x \right)}$:

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

            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:

               $$2 e^{2 x}$$

            Now plug in to the quotient rule:

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

         So, the result is: $10 \left(- 2 x e^{2 x} + e^{2 x}\right) e^{- 4 x}$

      So, the result is: $- 10 \left(- 2 x e^{2 x} + e^{2 x}\right) e^{- 4 x}$

   The result is: $- 10 \left(- 2 x e^{2 x} + e^{2 x}\right) e^{- 4 x} + 10 e^{2 x}$

2. Now simplify:

   $$10 \left(2 x + e^{4 x} - 1\right) e^{- 2 x}$$

The answer is:

$$10 \left(2 x + e^{4 x} - 1\right) e^{- 2 x}$$