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To find the derivative \( h'(x) \) of the function \( h(x) = (2 + f(\sin x))^3 \), we need to apply the **chain rule**.

### Step-by-Step Solution

1. **Outer function**: The outer function is \( (2 + f(\sin x))^3 \), so if we let \( u = 2 + f(\sin x) \), then \( h(x) = u^3 \).
   - The derivative of \( u^3 \) with respect to \( u \) is \( 3u^2 \).

2. **Inner function**: Now, differentiate \( u = 2 + f(\sin x) \) with respect to \( x \).
   - The derivative of \( 2 + f(\sin x) \) is:
     \[
     \frac{d}{dx}[2 + f(\sin x)] = f'(\sin x) \cdot \cos x
     \]
   - Here, \( f'(\sin x) \) is the derivative of \( f(\sin x) \) with respect to \( \sin x \), and then we multiply by the derivative of \( \sin x \), which is \( \cos x \).

3. **Applying the chain rule**: Now, combine these results using the chain rule:
   \[
   h'(x) = 3(2 + f(\sin x))^2 \cdot f'(\sin x) \cdot \cos x
   \]

### Identifying the Correct Answer

Looking at the provided choices, the correct expression for \( h'(x) \) matches:

- \[
\boxed{h'(x) = 3(2 + f(\sin x))^2 \cdot f'(\sin x) \cdot \cos x}
\]

This is the first choice in the provided options. 

### Explanation of Incorrect Choices

- The second option is incorrect because it only represents the derivative of the outer function, ignoring the chain rule.
- The third option incorrectly multiplies by \( \sin x \) instead of correctly applying the chain rule.
- The fourth option is incorrect because it misses the factor \( \cos x \), which is crucial due to the derivative of \( \sin x \).

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

Here are some related questions you might consider:
1. How do you find the derivative of nested functions using the chain rule?
2. What are some common mistakes when applying the chain rule?
3. Can you provide another example of differentiating a composite function?
4. What is the importance of correctly identifying inner and outer functions in differentiation?
5. How does the chain rule extend to functions of multiple variables?

**Tip:** When differentiating composite functions, always clearly identify the outer and inner functions to apply the chain rule correctly.

Let f be a differentiable function. If h(x) = (2 + f(sin(x)))^3, which of the following gives a correct process for finding h'(x)?

Find the derivative of the function h(x) = (2 + f(sin(x)))^3 using the chain rule. Learn how to apply the chain rule to differentiate composite functions, with a focus on trigonometric functions and their derivatives.

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