https://file.aiquickdraw.com/mathbot/file/1728814712273syd0lknt.jpg

Let's simplify the expression:

\[
f'(x) = \frac{2e^{2x} \cdot (8e^{2x} - 2e^x) - (4e^{2x} - 2e^x + 1) \cdot 4e^{2x}}{(2e^{2x})^2}
\]

### Step-by-Step Simplification:

1. **Simplify the denominator:**
   \[
   (2e^{2x})^2 = 4e^{4x}
   \]
   So the expression becomes:
   \[
   f'(x) = \frac{2e^{2x} \cdot (8e^{2x} - 2e^x) - (4e^{2x} - 2e^x + 1) \cdot 4e^{2x}}{4e^{4x}}
   \]

2. **Expand the terms in the numerator:**

   - For the first term: 
     \[
     2e^{2x} \cdot (8e^{2x} - 2e^x) = 2e^{2x} \cdot 8e^{2x} - 2e^{2x} \cdot 2e^x
     = 16e^{4x} - 4e^{3x}
     \]

   - For the second term: 
     \[
     (4e^{2x} - 2e^x + 1) \cdot 4e^{2x} = 4e^{2x} \cdot 4e^{2x} - 2e^x \cdot 4e^{2x} + 1 \cdot 4e^{2x}
     = 16e^{4x} - 8e^{3x} + 4e^{2x}
     \]

3. **Combine these two results:**
   The numerator is now:
   \[
   (16e^{4x} - 4e^{3x}) - (16e^{4x} - 8e^{3x} + 4e^{2x})
   \]
   Simplifying this:
   \[
   = 16e^{4x} - 4e^{3x} - 16e^{4x} + 8e^{3x} - 4e^{2x}
   \]
   Combine like terms:
   \[
   = (-4e^{3x} + 8e^{3x}) - 4e^{2x}
   = 4e^{3x} - 4e^{2x}
   \]

4. **Rewrite the entire expression:**
   \[
   f'(x) = \frac{4e^{3x} - 4e^{2x}}{4e^{4x}}
   \]

5. **Simplify the fraction:**
   Factor out the common term \( 4e^{2x} \) in the numerator:
   \[
   f'(x) = \frac{4e^{2x}(e^x - 1)}{4e^{4x}}
   \]
   Cancel out the \( 4e^{2x} \) terms:
   \[
   f'(x) = \frac{e^x - 1}{e^{2x}}
   \]

6. **Final simplified expression:**
   \[
   f'(x) = \frac{e^x - 1}{e^{2x}} = e^{-x} - e^{-2x}
   \]

### The simplified expression for \( f'(x) \) is:
\[
f'(x) = e^{-x} - e^{-2x}
\]

Would you like more details on any of the steps, or have any questions?

---

Here are some related questions for further exploration:

1. How would you apply the chain rule to differentiate exponential functions?
2. Can you verify the simplified derivative by differentiating the original function?
3. How does the behavior of \( e^{-x} - e^{-2x} \) change as \( x \to \infty \)?
4. What is the integral of \( f'(x) = e^{-x} - e^{-2x} \)?
5. How would you plot the function and its derivative to compare their behaviors?

**Tip:** When simplifying complex expressions, look for common factors in the numerator and denominator to make the process more efficient.

Simplify the given derivative expression.

Learn how to simplify a complex derivative involving exponential functions, step by step. This problem covers calculus concepts such as product rule, exponential function properties, and simplification of rational expressions. Ideal for students in Grades 11-12 or early undergraduate calculus.

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