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.