Suppose f and g are two differentiable functions. Some values of f(x), f
0
(x),
g(x) and g
0
(x) (for x = 2, 3, 5) are given in the table below.
x f(x) f
0
(x) g(x) g
0
(x)
2 10 −1 5 3
3 1 2 5 7
5 −3 4 −2 4
If h = f ◦ g is the composition h(x) = f(g(x)), what is the value of h
0
(3)?

To find the value of \( h'(3) \), where \( h(x) = f(g(x)) \), we need to use the chain rule for differentiation. The chain rule states that if \( h(x) = f(g(x)) \), then:

\[
h'(x) = f'(g(x)) \cdot g'(x)
\]

To determine \( h'(3) \), we substitute \( x = 3 \):

\[
h'(3) = f'(g(3)) \cdot g'(3)
\]

From the table, we have the following values:

- \( g(3) = 5 \)
- \( g'(3) = 7 \)

Now, we need \( f'(g(3)) = f'(5) \). From the table, \( f'(5) = 4 \).

Now, substitute these values into the expression for \( h'(3) \):

\[
h'(3) = f'(5) \cdot g'(3) = 4 \cdot 7 = 28
\]

Therefore, the value of \( h'(3) \) is:

\[
h'(3) = 28
\]

Would you like additional details or have questions?

Here are some related questions to explore further:

1. How does the chain rule apply to other compositions, like \( k(x) = g(f(x)) \)?
2. What changes if the functions \( f \) and \( g \) were not differentiable?
3. How would you interpret \( h(x) = f(g(x)) \) graphically?
4. What if \( g(x) \) were a constant function? How would \( h'(x) \) behave?
5. Can the chain rule be extended to compositions involving more than two functions?

**Tip:** For compositions, always differentiate the outer function evaluated at the inner function times the derivative of the inner function.

Suppose f and g are two differentiable functions. Some values of f(x), f'(x), g(x), and g'(x) (for x = 2, 3, 5) are given in the table below. If h = f ◦ g is the composition h(x) = f(g(x)), what is the value of h'(3)?

Explore how to find the derivative h'(3) for the composite function h(x) = f(g(x)) using the chain rule. This example includes a table of values for f(x), f'(x), g(x), and g'(x) and demonstrates how to apply the chain rule step-by-step. Perfect for students in Grades 11-12 studying calculus and differentiation.

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