To differentiate the function \(\frac{e^{2x} + e^x + 1}{e^x}\), we can 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. How do you differentiate a product of two functions?
2. What is the chain rule, and how is it applied in differentiation?
3. How would you integrate the function \(e^x - e^{-x}\)?
4. How do exponential functions differ from polynomial functions in terms of differentiation?
5. What are the applications of exponential functions in real-life problems?

**Tip:** Remember that simplifying an expression before differentiating often makes the differentiation process easier and less error-prone.

Learn how to differentiate the function (e^(2x) + e^x + 1) / e^x with a detailed step-by-step solution. This problem involves differentiation of exponential functions and application of the chain rule. Suitable for advanced high school students or those studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation