9. [-/1.07 Points] DETAILS MY NOTES SCALCET9 3.4.077. Let $$ r(x)=f(g(h(x))) $$, where $$ h(1)=4, g(4)=6, h^{\prime}(1)=4, g^{\prime}(4)=4 $$, and $$ f^{\prime}(6)=6 $$. Find $$ r^{\prime}(1) $$. $$ r^{\prime}(1)=\square $$

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We have $$ r(x) = f(g(h(x))) $$. To find $$ r'(x) $$, we use the chain rule:

$$
r'(x) = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)
$$

Evaluating at $$ x=1 $$:

1. Since $$ h(1)=4 $$, it follows that $$ g(h(1)) = g(4) $$.
2. Given $$ g(4)=6 $$, we have $$ f'(g( h(1) )) = f'(6) $$.
3. We are also given $$ h'(1)=4 $$ and $$ g'(4)=4 $$.
4. And $$ f'(6)=6 $$.

Thus,

$$
r'(1)= f'(6) \cdot g'(4) \cdot h'(1)= 6 \cdot 4 \cdot 4 = 96.
$$

The final answer is $$ r'(1)=96 $$.
$$ r'(1) = 96 $$

[-/1.07 Points] DETAILS MY NOTES SCALCET9 3.4.077.
Let , where , and . Find .

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[-/1.07 Points] DETAILS MY NOTES SCALCET9 3.4.077. Let , where , and . Find .