Suppose that f\!\left(\frac{\pi}{3}\right)=-2 and f'\!\left(\frac{\pi}{3}\right)=8, and let g(x) = f(x) \sin x and \displaystyle h(x) = \frac{\cos x}{f(x)}. Answer the following questions.

2. Find h'(\pi/3).
Answer: h'(\pi/3) =

We are given that:

- \( f\left( \frac{\pi}{3} \right) = -2 \)
- \( f'\left( \frac{\pi}{3} \right) = 8 \)
- \( g(x) = f(x) \sin x \)
- \( h(x) = \frac{\cos x}{f(x)} \)

We need to find \( h' \left( \frac{\pi}{3} \right) \).

### Step 1: Differentiate \( h(x) = \frac{\cos x}{f(x)} \)
We will use the quotient rule to differentiate \( h(x) \). The quotient rule states that:

\[
h'(x) = \frac{d}{dx} \left( \frac{\cos x}{f(x)} \right) = \frac{f(x) \cdot (-\sin x) - \cos x \cdot f'(x)}{[f(x)]^2}
\]

### Step 2: Plug in the given values at \( x = \frac{\pi}{3} \)
At \( x = \frac{\pi}{3} \), we know the following values:

- \( \cos \left( \frac{\pi}{3} \right) = \frac{1}{2} \)
- \( \sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} \)
- \( f \left( \frac{\pi}{3} \right) = -2 \)
- \( f' \left( \frac{\pi}{3} \right) = 8 \)

Substitute these values into the derivative formula:

\[
h' \left( \frac{\pi}{3} \right) = \frac{-2 \cdot \frac{\sqrt{3}}{2} - \frac{1}{2} \cdot 8}{(-2)^2}
\]

### Step 3: Simplify the expression

\[
h' \left( \frac{\pi}{3} \right) = \frac{-\sqrt{3} - 4}{4}
\]

\[
h' \left( \frac{\pi}{3} \right) = \frac{-\sqrt{3} - 4}{4}
\]

Thus, the final result is:

\[
h' \left( \frac{\pi}{3} \right) = \frac{-\sqrt{3} - 4}{4}
\]

### Answer:
\[
h' \left( \frac{\pi}{3} \right) = \frac{-\sqrt{3} - 4}{4}
\]

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Do you want details on any step or have questions?  
Here are some related questions to explore:

1. How would you apply the chain rule in a more complex function?
2. Can you find \( g' \left( \frac{\pi}{3} \right) \) given \( g(x) = f(x) \sin x \)?
3. What happens to the quotient rule when both functions have the same value at a point?
4. How would the solution change if \( f' \left( \frac{\pi}{3} \right) \) were negative?
5. Can you explain the significance of using exact values versus approximations for trigonometric functions?

**Tip**: When applying the quotient rule, always check if simplification is possible before plugging in values to make calculations easier.

Suppose that f(π/3)=-2 and f'(π/3)=8, and let g(x) = f(x) sin(x) and h(x) = cos(x)/f(x). Find h'(π/3).

Learn how to differentiate h(x) = cos(x)/f(x) at x = π/3 using the quotient rule. This calculus problem includes detailed steps, key trigonometric values, and the application of the quotient rule. Suitable for advanced high school students in Grades 11-12.

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