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. Identify the functions:
   - $$ g(x)=8\ln(x) $$
   - $$ h(x)=\sin(x) $$

2. Compute the derivatives:
   - $$ g'(x)=\frac{8}{x} $$
   - $$ h'(x)=\cos(x) $$

3. Apply the Chain Rule:
   - Since $$ F(x)=h(g(x)) $$, then by the Chain Rule,
     $$
     F^{\prime}(x)=h^{\prime}(g(x))\cdot g^{\prime}(x)
     $$

4. Substitute the given functions and derivatives:
   - Replace $$ h^{\prime}(g(x)) $$ with $$ \cos(g(x)) $$:
     $$
     h^{\prime}(g(x))=\cos(8\ln(x))
     $$
   - Thus,
     $$
     F^{\prime}(x)=\cos(8\ln(x))\cdot\frac{8}{x}
     $$

5. Write the final answer:
   $$
   F^{\prime}(x)=\frac{8\cos(8\ln(x))}{x}
   $$
$$ F^{\prime}(x) = \frac{8 \cos(8 \ln(x))}{x} $$

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