https://file.aiquickdraw.com/mathbot/file/1726192023636cdulvvss.jpg

The function \( f(x) \) given in the image is:

\[
f(x) = \frac{1}{(e x)^2} + e^{\pi}
\]

This can be simplified and analyzed as follows:

### Step-by-step Breakdown:

1. **Simplification**:
   \[
   f(x) = \frac{1}{(e^2 x^2)} + e^{\pi}
   \]
   Where \( e^2 \) is Euler's number squared, and \( e^{\pi} \) is a constant term.

2. **Derivative**: To compute the derivative \( f'(x) \), we differentiate the two parts of the function separately:
   - For \( \frac{1}{(e^2 x^2)} \), use the chain rule:
     \[
     \frac{d}{dx} \left( \frac{1}{(e^2 x^2)} \right) = -\frac{2}{e^2 x^3}
     \]
   - The term \( e^{\pi} \) is constant, so its derivative is zero.

Thus, the derivative is:
\[
f'(x) = -\frac{2}{e^2 x^3}
\]

3. **Behavior**:
   - For large values of \( x \), \( f(x) \) tends to \( e^{\pi} \), since \( \frac{1}{e^2 x^2} \) approaches zero.
   - For small values of \( x \), \( f(x) \) becomes very large because of the \( \frac{1}{e^2 x^2} \) term.

Would you like to explore further properties of this function, such as its critical points or limits?

### Related Questions:
1. What is the second derivative of \( f(x) \)?
2. How does \( f(x) \) behave as \( x \to 0 \)?
3. What is the limit of \( f(x) \) as \( x \to \infty \)?
4. How can we approximate \( e^{\pi} \)?
5. How would we find the critical points of \( f(x) \)?

### Tip:
When simplifying expressions with constants like \( e \) or \( \pi \), always treat them as specific numbers for easier computation and interpretation.

What is the derivative of the function f(x) = 1/(ex)^2 + e^π?

Explore how to find the derivative of f(x) = 1/(ex)^2 + e^π using the chain rule. This problem involves calculus concepts such as differentiation and simplification of expressions. Suitable for students in Grades 11-12.

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