To differentiate the function $\frac{e^{2x} - e^x + 1}{e^x}$, let's start by simplifying the expression.

Step 1: Simplify the Expression

$\frac{e^{2x} - e^x + 1}{e^x}$

We can rewrite each term in the numerator divided by $e^x$:

$= \frac{e^{2x}}{e^x} - \frac{e^x}{e^x} + \frac{1}{e^x}$

$= e^{2x - x} - e^{x - x} + e^{-x}$

$= e^x - 1 + e^{-x}$

Step 2: Differentiate the Simplified Expression

Now we need to differentiate $e^x - 1 + e^{-x}$ with respect to $x$:

1. The derivative of $e^x$ is $e^x$.
2. The derivative of a constant, like $-1$, is $0$.
3. The derivative of $e^{-x}$ is $-e^{-x}$ (using the chain rule).

Putting it all together:

$\frac{d}{dx}\left(e^x - 1 + e^{-x}\right) = e^x + 0 - e^{-x}$

$= e^x - e^{-x}$

Final Answer

$\frac{d}{dx}\left(\frac{e^{2x} - e^x + 1}{e^x}\right) = e^x - e^{-x}$

Would you like more details on any part of this process or have any questions? Here are some related questions you might find interesting:

1. What is the derivative of $\frac{e^{2x} + e^x + 1}{e^x}$?
2. How do you apply the quotient rule to differentiate functions?
3. What is the integral of $e^x - e^{-x}$?
4. How does the chain rule work in differentiation?
5. How do exponential functions relate to their derivatives?

Tip: Always consider simplifying an expression before differentiating. It can save time and reduce the chance of making mistakes.