If y = sin x cos x, then at x = pi/3, dy/dx = ?

a. -1/2 
b. -1/4 
c. 1/4 
d. 1/2 
e. 1

We are given the function \( y = \sin x \cos x \), and we need to find the derivative \( \frac{dy}{dx} \) at \( x = \frac{\pi}{3} \).

### Step 1: Differentiate \( y = \sin x \cos x \)

We can apply the **product rule** for differentiation here. The product rule states that if \( y = u(x) v(x) \), then:

\[
\frac{dy}{dx} = u'(x) v(x) + u(x) v'(x)
\]

In this case:
- \( u(x) = \sin x \), so \( u'(x) = \cos x \)
- \( v(x) = \cos x \), so \( v'(x) = -\sin x \)

Now, applying the product rule:

\[
\frac{dy}{dx} = \cos x \cdot \cos x + \sin x \cdot (-\sin x)
\]
\[
\frac{dy}{dx} = \cos^2 x - \sin^2 x
\]

### Step 2: Substitute \( x = \frac{\pi}{3} \)

We now substitute \( x = \frac{\pi}{3} \) into the derivative.

- \( \cos \frac{\pi}{3} = \frac{1}{2} \)
- \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \)

Substituting these values into the expression for \( \frac{dy}{dx} \):

\[
\frac{dy}{dx} = \left( \frac{1}{2} \right)^2 - \left( \frac{\sqrt{3}}{2} \right)^2
\]
\[
\frac{dy}{dx} = \frac{1}{4} - \frac{3}{4}
\]
\[
\frac{dy}{dx} = \frac{-2}{4} = -\frac{1}{2}
\]

### Conclusion:

At \( x = \frac{\pi}{3} \), the derivative \( \frac{dy}{dx} = -\frac{1}{2} \).

Thus, the correct answer is:
\[
\boxed{-\frac{1}{2}} \quad \text{(option a)}
\]

Would you like further details on any part of the solution or have any questions?

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Here are some related questions you might find useful:
1. What is the derivative of \( \sin^2 x \)?
2. How does the product rule differ from the chain rule in differentiation?
3. Can you prove the identity \( \cos^2 x - \sin^2 x = \cos 2x \)?
4. What is the second derivative of \( y = \sin x \cos x \)?
5. How would you integrate \( \sin x \cos x \)?

**Tip:** For trigonometric products, it’s often useful to express them using trigonometric identities to simplify differentiation or integration.

This problem focuses on finding the derivative of the function y = sin x cos x using the product rule and trigonometric identities. Learn the step-by-step solution for differentiating at x = pi/3 and apply key concepts in calculus. Suitable for Grades 11-12.

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