To find \( g'(3) \), we need to use the chain rule, given that \( g(x) = h(x^2) \).

### Step-by-Step Solution:

1. **Express \( g'(x) \) in terms of \( h'(x) \):**
   \[
   g(x) = h(x^2)
   \]
   Differentiate both sides with respect to \( x \):
   \[
   g'(x) = \frac{d}{dx} \left[h(x^2)\right]
   \]
   Apply the chain rule:
   \[
   g'(x) = h'(x^2) \cdot \frac{d}{dx}(x^2)
   \]

2. **Differentiate \( x^2 \):**
   \[
   \frac{d}{dx}(x^2) = 2x
   \]

3. **Substitute back into the expression for \( g'(x) \):**
   \[
   g'(x) = h'(x^2) \cdot 2x
   \]

4. **Evaluate \( g'(3) \):**
   Substitute \( x = 3 \) into the expression:
   \[
   g'(3) = h'((3)^2) \cdot 2 \cdot 3
   \]
   Simplify:
   \[
   g'(3) = h'(9) \cdot 6
   \]

5. **Substitute the given value \( h'(9) = 17 \):**
   \[
   g'(3) = 17 \cdot 6 = 102
   \]

### Final Answer:
\[
g'(3) = 102
\]

Would you like further details or have any questions about this solution?

Here are some related questions:

1. How would the solution change if \( h'(9) \) were a different value?
2. What is the general formula for \( g'(x) \) in terms of \( h'(x) \)?
3. Can you explain the chain rule and how it was applied here?
4. What would \( g'(x) \) be if \( g(x) \) were instead \( h(x^3) \)?
5. How would you interpret \( g(x) = h(x^2) \) graphically?

**Tip:** Always check if a function inside another function can be simplified before applying differentiation rules.

Discover how to find g'(3) using the chain rule and differentiation. Step-by-step solution provided for g(x) = h(x^2), with h'(9) = 17. Learn the application of differentiation rules in calculus.

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