https://file.aiquickdraw.com/mathbot/file/17286189239644h5o9p72.jpg

To solve the problem given in the image, we'll use the chain rule for derivatives. Let's go through each part one by one.

### Part (a)
We are given that:
\[ h(x) = f(g(x)) \]
We need to find \( h'(1) \).

Using the chain rule:
\[ h'(x) = f'(g(x)) \cdot g'(x) \]
So, to find \( h'(1) \):
\[ h'(1) = f'(g(1)) \cdot g'(1) \]

From the table:
- \( g(1) = 2 \)
- \( g'(1) = 6 \)
- \( f'(2) = 5 \)

Therefore:
\[ h'(1) = f'(2) \cdot g'(1) = 5 \cdot 6 = 30 \]

### Part (b)
We are given that:
\[ H(x) = g(f(x)) \]
We need to find \( H'(2) \).

Using the chain rule:
\[ H'(x) = g'(f(x)) \cdot f'(x) \]
So, to find \( H'(2) \):
\[ H'(2) = g'(f(2)) \cdot f'(2) \]

From the table:
- \( f(2) = 1 \)
- \( f'(2) = 5 \)
- \( g'(1) = 6 \)

Therefore:
\[ H'(2) = g'(1) \cdot f'(2) = 6 \cdot 5 = 30 \]

### Answers
- (a) \( h'(1) = 30 \)
- (b) \( H'(2) = 30 \)

Would you like more details or have any questions?

Here are some related questions to expand your understanding:
1. How do we use the chain rule in general for composite functions?
2. What happens if the functions are not differentiable at the points provided?
3. How does the value of \( f(x) \) affect the derivative calculation when composing functions?
4. Can you think of a real-world scenario where chain rule might be used?
5. How would you find the second derivative for the function \( h(x) = f(g(x)) \)?

**Tip:** Remember, the chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

Given the differentiable functions f and g, with their values and derivatives for various x, solve the following:
(a) If h(x) = f(g(x)), find h'(1).
(b) If H(x) = g(f(x)), find H'(2).

Learn how to solve a calculus problem using the chain rule for derivatives. This example involves finding the derivatives of composite functions h(x) = f(g(x)) and H(x) = g(f(x)) at specific points using given values and their derivatives. Suitable for students in Grades 11-12.

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