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

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

         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: $-3$

         The result of the chain rule is:

         $$- 3 e^{- 3 x}$$

      So, the result is: $- 6 e^{- 3 x}$

   2. 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: $18 e^{3 x}$

   The result is: $18 e^{3 x} - 6 e^{- 3 x}$

2. Now simplify:

   $$6 \cdot \left(3 e^{6 x} - 1\right) e^{- 3 x}$$

The answer is:

$$6 \cdot \left(3 e^{6 x} - 1\right) e^{- 3 x}$$