We have determined that $$ g^{\prime}(x)=\frac{8}{x} $$ and $$ h^{\prime}(x)=\cos (x) $$, where $$ g(x)=8 \ln (x) $$ and $$ h(x)=\sin (x) $$. Applying the Chain Rule, gives the following result. $$ \begin{array}{l} F(x)=h(g(x)) \ F^{\prime}(x)=h^{\prime}(g(x)) \cdot g^{\prime}(x) \ = \ ext { SUBMIT SKIP (YOU CANNOT COME BACK) } \ ext { Need Help? Read It }\end{array} $$ SUBNIT ANSWER

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We have determined that $$ g^{\prime}(x)=\frac{8}{x} $$ and $$ h^{\prime}(x)=\cos (x) $$, where $$ g(x)=8 \ln (x) $$ and $$ h(x)=\sin (x) $$. Applying the Chain Rule, gives the following result. $$ \begin{array}{l} F(x)=h(g(x)) \ F^{\prime}(x)=h^{\prime}(g(x)) \cdot g^{\prime}(x) \ = \ 	ext { SUBMIT SKIP (YOU CANNOT COME BACK) } \ 	ext { Need Help? Read It }\end{array} $$ SUBNIT ANSWER

UpStudy AI · Accepted Answer

1. We start with the composite function:
   
   $$
   F(x)=h(g(x))=\sin(8\ln(x))
   $$

2. The Chain Rule states that if $$ F(x)=h(g(x)) $$, then:

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

3. Given that:
   
   $$
   g(x)=8\ln(x) \quad \text{and} \quad g'(x)=\frac{8}{x}
   $$
   
   $$
   h(x)=\sin(x) \quad \text{and} \quad h'(x)=\cos(x)
   $$

4. Applying the Chain Rule, we substitute into the formula:
   
   $$
   F'(x)=\cos(g(x))\cdot g'(x)=\cos(8\ln(x))\cdot\frac{8}{x}
   $$

5. Therefore, the derivative is:
   
   $$
   F'(x)=\frac{8}{x}\cos(8\ln(x))
   $$
$$ F'(x) = \frac{8}{x} \cos(8 \ln(x)) $$

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