Derivative of exp(3*x)+2*exp(-x)

1. Differentiate $e^{3 x} + 2 e^{- x}$ term by term:

   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:

      $$3 e^{3 x}$$

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

      1. Let $u = - x$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- x\right)$:

         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: $-1$

         The result of the chain rule is:

         $$- e^{- x}$$

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

   The result is: $3 e^{3 x} - 2 e^{- x}$

2. Now simplify:

   $$\left(3 e^{4 x} - 2\right) e^{- x}$$

The answer is:

$$\left(3 e^{4 x} - 2\right) e^{- x}$$