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To solve the given problem, let’s carefully work through the information provided in the image:

The problem is:

**If \( f(x) = 5 \cos^2(\pi - x) \), then \( f'\left(\frac{\pi}{2}\right) \) is?**

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### Step-by-step Solution:

1. **Rewrite \( f(x) \):**
   Using the identity for cosine:
   \[
   \cos(\pi - x) = -\cos(x)
   \]
   Thus:
   \[
   f(x) = 5 \cos^2(\pi - x) = 5 \cos^2(-x)
   \]
   And since \( \cos(-x) = \cos(x) \):
   \[
   f(x) = 5 \cos^2(x)
   \]

2. **Differentiate \( f(x) \):**
   Using the chain rule:
   \[
   \frac{d}{dx}[\cos^2(x)] = 2\cos(x)(-\sin(x)) = -2\cos(x)\sin(x)
   \]
   So:
   \[
   f'(x) = 5 \cdot (-2\cos(x)\sin(x)) = -10\cos(x)\sin(x)
   \]

3. **Simplify \( f'(x) \):**
   Using the double-angle formula:
   \[
   \sin(2x) = 2\sin(x)\cos(x)
   \]
   Then:
   \[
   f'(x) = -10 \cdot \frac{\sin(2x)}{2} = -5\sin(2x)
   \]

4. **Evaluate \( f'\left(\frac{\pi}{2}\right) \):**
   Substitute \( x = \frac{\pi}{2} \) into \( f'(x) = -5\sin(2x) \):
   \[
   f'\left(\frac{\pi}{2}\right) = -5\sin\left(2 \cdot \frac{\pi}{2}\right) = -5\sin(\pi)
   \]
   Since \( \sin(\pi) = 0 \):
   \[
   f'\left(\frac{\pi}{2}\right) = -5 \cdot 0 = 0
   \]

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### Final Answer:
\[
\boxed{0}
\]

Let me know if you want more details or have any questions! Here are some related questions to explore:

1. What are the key trigonometric identities used in differentiating trigonometric functions?
2. How does the chain rule apply to composite functions like \( \cos^2(x) \)?
3. What is the significance of the double-angle formulas in simplifying trigonometric derivatives?
4. How does \( f(x) = \cos^2(x) \) change when evaluated over different intervals?
5. Why is \( \sin(\pi) \) equal to zero?

**Tip:** Always simplify trigonometric expressions using standard identities before differentiating to make calculations easier.

If f(x) = 5cos²(π - x), then f'(π/2) is?

Solve the problem of differentiating f(x) = 5cos²(π - x) to find f'(π/2). Learn key trigonometric identities and differentiation techniques, including the chain rule and double-angle formula. Suitable for Grades 11-12 or early college students.

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