To find the derivative of the function $f(x) = e^{2x} + e^{-x}$, we can differentiate each term separately using the rules of differentiation.

Step-by-Step Process:

1. Differentiate the first term: $e^{2x}$:

   • The derivative of $e^{2x}$ is $2e^{2x}$, because of the chain rule (the derivative of $2x$ is $2$).

2. Differentiate the second term: $e^{-x}$:

   • The derivative of $e^{-x}$ is $-e^{-x}$, again using the chain rule (the derivative of $-x$ is $-1$).

Putting it together:

$f'(x) = 2e^{2x} - e^{-x}$

So, the derivative of the function is: $f'(x) = 2e^{2x} - e^{-x}$

Would you like further details on the chain rule or any other part of this solution?

Related Questions:

1. What is the second derivative of $f(x) = e^{2x} + e^{-x}$?
2. How do we apply the chain rule in finding derivatives?
3. What is the general form of the derivative of $e^{ax}$?
4. Can you integrate $2e^{2x} - e^{-x}$ back to find the original function?
5. How does the behavior of $f(x) = e^{2x} + e^{-x}$ change as $x \to \infty$?

Tip:

When differentiating exponential functions, always pay attention to the inner functions, as the chain rule often applies.