Compute and simplify the difference quotient \( \frac{f(x+h)-f(x)}{h}, h \neq 0 \), for \( f(x)=\frac{1}{x+25} \) A. \( \frac{1}{(x+h+25)(x+25)} \) B. \( -\frac{25}{(x+h+25)(x+25)} \) D. \( -\frac{1}{(x+h+25)(x+25)} \) \( (x+1)^{2} \)

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

\( \left. \begin{array} { l } { y = \cos ^ { 3 } ( 5 + 8 x ) } \\ { y ^ { \prime } - ? } \end{array} \right. \)

Find the area bounded by the graphs of the indicated equations over the given interval. \[ y=-x^{2}+29 ; y=0 ;-3 \leq x \leq 3 \] The area is \( \square \) square units.

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