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To solve the given problem, we need to evaluate two compositions of functions and their derivatives:

1. **Given Information:**
   - \( f(7) = 29 \)
   - \( f'(7) = 4 \)
   - \( f(29) = 12 \)
   - \( f'(29) = -6 \)
   - \( g(29) = -6 \)
   - \( g'(29) = -10 \)
   - \( g(12) = -27 \)
   - \( g'(12) = 42 \)

2. **(a) Find \( (g \circ f)'(29) \):**

   We need to find the derivative of the composite function \( (g \circ f)(x) \) at \( x = 29 \).

   The chain rule for the derivative of a composite function \( (g \circ f)(x) \) is:
   \[
   (g \circ f)'(x) = g'(f(x)) \cdot f'(x)
   \]

   So, we need to find \( g'(f(29)) \) and \( f'(29) \).

   - From the given information, \( f(29) = 12 \).
   - We also know \( f'(29) = -6 \).
   - Next, we need \( g'(12) \). From the given information, \( g'(12) = 42 \).

   Now, substituting into the chain rule:
   \[
   (g \circ f)'(29) = g'(12) \cdot f'(29) = 42 \cdot (-6) = -252
   \]

   Therefore, \( (g \circ f)'(29) = -252 \).

3. **(b) Find \( (f \circ g)'(7) \):**

   We need to find the derivative of the composite function \( (f \circ g)(x) \) at \( x = 7 \).

   Using the chain rule for the derivative of \( (f \circ g)(x) \):
   \[
   (f \circ g)'(x) = f'(g(x)) \cdot g'(x)
   \]

   So, we need to find \( f'(g(7)) \) and \( g'(7) \).

   - First, we need \( g(7) \), but this is not directly provided in the given information. Let's review what we have:
   - There is no value of \( g(7) \) given directly.

   Given the missing value of \( g(7) \), it seems the problem might have a typo or missing data. Without \( g(7) \), we cannot calculate \( f'(g(7)) \). If this information is available, we can proceed with finding \( f'(g(7)) \cdot g'(7) \).

**Conclusion:**

- **(a)** \( (g \circ f)'(29) = -252 \).
- **(b)** For \( (f \circ g)'(7) \), we need \( g(7) \) to continue.

Would you like more details, or do you have any questions?

**Follow-up Questions:**
1. How does the chain rule apply to other composite functions?
2. What happens if one of the function's derivatives is zero at the point of interest?
3. Can we generalize this rule to functions of more than one variable?
4. What are common pitfalls when applying the chain rule?
5. How does the chain rule extend to second derivatives?

**Tip:** Always ensure all necessary function values and derivatives are known before attempting to evaluate a composite derivative.

Let f and g be functions such that f(7)=29, f'(7)=4, f(29)=12, f'(29)=-6, g(29)=-6, g'(29)=-10, g(12)=-27, and g'(12)=42. Find (a) (g ∘ f)'(29) and (b) (f ∘ g)'(7).

This problem involves applying the chain rule to find the derivative of composite functions f and g. Given function values and derivatives at specific points, solve for (g ∘ f)'(29) and (f ∘ g)'(7).

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