19. [-/1 Points] DETAILS MY NOTES SCALCET9 3.4.067. If $$ F(x)=f(g(x)) $$, where $$ f(-3)=5, f^{\prime}(-3)=2, f^{\prime}(2)=3, g(2)=-3 $$, and $$ g^{\prime}(2)=7 $$, find $$ F^{\prime}(2) $$. $$ F^{\prime}(2)=\square $$

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UpStudy AI · Accepted Answer

We are given $$ F(x) = f(g(x)) $$ and the chain rule tells us that

$$
F'(x) = f'(g(x)) \cdot g'(x)
$$

Substituting $$ x = 2 $$:

$$
F'(2) = f'(g(2)) \cdot g'(2)
$$

Given that $$ g(2) = -3 $$ and $$ g'(2) = 7 $$, we have

$$
F'(2) = f'(-3) \cdot 7
$$

Since $$ f'(-3) = 2 $$, it follows that

$$
F'(2) = 2 \cdot 7 = 14
$$

Thus, the final answer is

$$
F'(2)=14
$$
$$ F'(2) = 14 $$

[-/1 Points] DETAILS MY NOTES SCALCET9 3.4.067.
If , where , and , find .

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[-/1 Points] DETAILS MY NOTES SCALCET9 3.4.067. If , where , and , find .