Suppose that the functions \( f \) and \( g \) are defined as follows. \[ \begin{array}{l}f(x)=\frac{5}{x}, x \neq 0 \\ g(x)=x^{2}-4\end{array} \] Find the compositions \( f \) of and \( g \circ g \).

Calculate \( \int_{9}^{7} 3 x^{2} d x \), given the following. \( \int_{4}^{7} x d x=16.5 \int_{4}^{7} x^{2} d x=93 \int_{7}^{9} x^{2} d x=\frac{386}{3} \) \( \int_{9}^{7} 3 x^{2} d x=\square \) (Type an integer or a simplified fraction.)

Find \( C_{x y}(x, y) \) if \( C(x, y)=8 x^{2}+3 x y-9 y^{2}+3 x-3 y-3 \) \( C_{x y}(x, y)=\square \)

Find \( f^{\prime}(x) \). \( f(x)=\left(3 x^{5}+4\right)^{3} \) \( f^{\prime}(x)=\square \)

Use L'Hôpital's rule to find the following limit and check the answer using an algebraic simplification. \( \lim _{x \rightarrow 8} \frac{x^{2}-x-56}{x-8} \) \( \lim _{x \rightarrow 8} \frac{x^{2}-x-56}{x-8}=\square \) (Simplify your answer.)

Find \( (A) f^{\prime}(x) \), (B) the partition numbers for \( f^{\prime} \), and \( (C) \) the critical numbers of \( f \). \( f(x)=x^{3}-3 x+7 \) (A) \( f^{\prime}(x)= \) (B) Find the partition numbers for \( f^{\prime} \). Select the correct choice below and, if necessary, fill in the answer box to cor A. The partition pumber(s) is/are \( x=\square \). There are no partition numbers. A. The critical number(s) is/are \( x=\square \). (C) Find the critical numbers for \( f \). Select the correct choice below and, if necessary, fill in the answer box to comp (Use a comma to separate answers as needed.) B. There are no critical numbers.