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The problem asks for the value of \( g'(7) \) given that \( h(g(x)) = x \) for all \( x \). We are also provided with selected values for \( h(x) \) and \( h'(x) \) from the table.

We can solve this using the **chain rule**. 

### Step 1: Differentiate the equation
Given that \( h(g(x)) = x \), we differentiate both sides with respect to \( x \):
\[
\frac{d}{dx} [h(g(x))] = \frac{d}{dx} [x]
\]
By the chain rule:
\[
h'(g(x)) \cdot g'(x) = 1
\]
Thus, we have the relation:
\[
g'(x) = \frac{1}{h'(g(x))}
\]

### Step 2: Apply to \( g'(7) \)
We need to find \( g'(7) \). From the table, we see that when \( x = 7 \), \( h(7) = 22 \) and \( h'(7) = 10 \).

Using the equation for \( g'(x) \), we substitute \( x = 7 \):
\[
g'(7) = \frac{1}{h'(g(7))}
\]

Now, since \( h(g(7)) = 7 \), we know \( g(7) = 3 \) (from the table where \( h(3) = 7 \)).

Thus:
\[
g'(7) = \frac{1}{h'(3)}
\]

### Step 3: Use the value of \( h'(3) \)
From the table, we know that \( h'(3) = 5 \). Therefore:
\[
g'(7) = \frac{1}{5}
\]

### Final Answer:
The value of \( g'(7) \) is \( \frac{1}{5} \), which corresponds to option **(C)**.

Would you like further clarification or details on any steps? Here are five related questions you might find useful:

1. How does the chain rule apply to composite functions like \( h(g(x)) \)?
2. What other methods can be used to differentiate implicitly?
3. How can the derivative of an inverse function be found using similar methods?
4. Why is \( g(7) = 3 \) and not a different value based on the table?
5. How would you approach solving if the table provided different values for \( h(x) \)?

**Tip:** The chain rule is essential for handling derivatives of compositions of functions, especially when inverse functions are involved.

Selected values of the increasing function h and its derivative h' are shown in the table above. If g is a differentiable function such that h(g(x)) = x for all x, what is the value of g'(7)?

In this calculus problem, learn how to use the chain rule to find the derivative of an inverse function. Given that h(g(x)) = x, and using a table with values for h(x) and h'(x), calculate the value of g'(7). Suitable for high school or college-level calculus students.

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