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The problem asks whether the derivative of \(\cos(x)\) at \(x = \frac{\pi}{2}\) is positive or negative.

### Solution:
1. The derivative of \(\cos(x)\) is:
   \[
   \frac{d}{dx} \cos(x) = -\sin(x)
   \]

2. Now, evaluate \(-\sin(x)\) at \(x = \frac{\pi}{2}\):
   \[
   -\sin\left(\frac{\pi}{2}\right) = -1
   \]

Since \(-1\) is negative, the derivative of \(\cos(x)\) at \(x = \frac{\pi}{2}\) is **negative**.

Would you like further details or clarifications?

### Follow-up Questions:
1. What is the derivative of \(\sin(x)\) at \(x = \frac{\pi}{2}\)?
2. How does the graph of \(\cos(x)\) behave near \(x = \frac{\pi}{2}\)?
3. Can you explain why the derivative of \(\cos(x)\) is \(-\sin(x)\)?
4. What is the second derivative of \(\cos(x)\)?
5. How does the derivative of \(\sin(x)\) compare to the derivative of \(\cos(x)\)?

### Tip:
For trigonometric functions, always remember to differentiate based on known rules: \(\frac{d}{dx} \sin(x) = \cos(x)\) and \(\frac{d}{dx} \cos(x) = -\sin(x)\).

Is the derivative of cos(x) at x = π/2 positive or negative?

Explore whether the derivative of cos(x) at x = π/2 is positive or negative. This calculus problem involves the differentiation of trigonometric functions. Learn how the formula for the derivative of cos(x) results in a negative value at x = π/2. Suitable for students in Grades 11-12.

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