5. (5 pts) Given that \( f(x, y, z)=x y+y z-x z, P(1,1,-1) \) and \( \mathbf{v}=\langle 1,-2,4\rangle \), find the directional derivative of \( f, D_{\rightarrow} f(P) \), in the direction of \( \mathbf{v} \).

1) \( \int_{8}^{\frac{\text { III }}{\frac{\pi}{2}}} x \cdot c \)

Integrate as indicated. Assume \( \mathrm{g}(\mathrm{x})>0 \) whenever \( \ln g(\mathrm{x}) \) is involved. \( \int \frac{8 \mathrm{x}}{4 \mathrm{x}^{2}+1} \mathrm{dx} \) \( \int \frac{8 \mathrm{x}}{4 \mathrm{x}^{2}+1} \mathrm{dx}= \)

\( \lim _ { x \rightarrow \infty } ( \frac { 3 x + 1 } { 3 x - 5 } ) ^ { \frac { 3 x ^ { 5 } + 4 x - 3 } { ( 2 x - 1 ) ^ { 2 } ( x + 2 ) ^ { 2 } } } \)

1. \( \lim _{x \rightarrow \infty}(\overline{3 x-5}) \) 2. \( \lim _{x \rightarrow 0}(1-\tan x)^{\frac{4 x}{\sin ^{2} 3 x}} \)

Find \( (A) f^{\prime}(x) \), (B) the partition numbers for \( f^{\prime} \), and \( (C) \) the critical numbers of \( f \). \( f(x)=x^{3}-12 x-8 \) (A) \( f^{\prime}(x)=\square \) (B) Find the partition numbers for \( f^{\prime} \). Select the correct choice below and, if necessary, fill in the answer box to complete your choic A. The partition pumber(s) is/are \( x=\square \). (Use a comma to separate answers as needed.) B. There are no partition numbers. (C) Find the critical numbers for \( f \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The critical number(s) is/are \( x=\square \). (Use a comma to separate answers as needed.)