Step 1 Recall the Chain Rule in terms of the functions \( h(x) \) and \( g(x) \), which states that if \( g \) is differentiable at \( x \) and \( h \) is differentiable at \( g(x) \), then the composite function \( F=h \) o \( g \) defined by \( F(x)=h(g(x)) \) is differentiable at \( x \) and \( F^{\prime} \) is given by the following. \[ F^{\prime}(x)=h^{\prime}(g(x)) \cdot g^{\prime}(x) \] We are given the function \( f(x)=\sin (8 \ln (x)) \) To write this is the form \( F(x)=h(g(x)) \) we can let \( g(x)=8 \ln (x) \) and \( h(x)= \) SUBMI SUB (YOU CANNOT COME BACK) Need Help? Read It SUBMIT ANSWER SKI

To express the function \( f(x) = \sin(8 \ln(x)) \) in the form \( F(x) = h(g(x)) \), we indeed set \( g(x) = 8 \ln(x) \). Now, we need to find the function \( h(x) \). Notice that \( h(x) \) should be the function that takes the output of \( g(x) \) and provides us with \( f(x) \). Therefore, if we let \( h(x) = \sin(x) \), then we can express \( f(x) \) as \( F(x) = h(g(x)) = h(8 \ln(x)) = \sin(8 \ln(x)) \). Don’t forget to check how the chain rule applies in this context! First, find the derivatives of both \( h \) and \( g \): - The derivative \( h'(x) = \cos(x) \). - The derivative \( g'(x) = \frac{8}{x} \). Now you can easily compute \( F'(x) \) using the chain rule! Doing derivatives in calculus might seem a bit mystifying at first, but it’s ultimately about understanding how functions interact with one another! Think of it like following a recipe: you have your main ingredients (the functions) and you have to mix them together (taking derivatives) to create something delicious (the derivative of the composite function).