Derivative of 3e^(2x)+4e^x

1. Differentiate $4 e^{x} + 3 e^{2 x}$ 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: $6 e^{2 x}$

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

      1. The derivative of $e^{x}$ is itself.

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

   The result is: $6 e^{2 x} + 4 e^{x}$

2. Now simplify:

   $$\left(6 e^{x} + 4\right) e^{x}$$

The answer is:

$$\left(6 e^{x} + 4\right) e^{x}$$