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. \( \begin{array}{l} F(x)=h(g(x)) \\ F^{\prime}(x)=h^{\prime}(g(x)) \cdot g^{\prime}(x) \\ = \\ \text { SUBMIT SKIP (YOU CANNOT COME BACK) } \\ \text { Need Help? Read It }\end{array} \) SUBNIT ANSWER
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To apply the Chain Rule for \( F(x) = h(g(x)) \), we start with what we know: 1. We have \( h'(x) = \cos(x) \) and \( g'(x) = \frac{8}{x} \). 2. Plugging \( g(x) \) into \( h' \), we get \( h'(g(x)) = \cos(g(x)) \). Now, substituting into the Chain Rule formula: \[ F'(x) = h'(g(x)) \cdot g'(x) = \cos(g(x)) \cdot \frac{8}{x}. \] Finally, remember that \( g(x) = 8\ln(x) \), therefore: \[ F'(x) = \cos(8\ln(x)) \cdot \frac{8}{x}. \] And there you have your differentiated function!
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