for the function below, find (a) ( f^{prime}(x) ), (b) the partition numbers for ( f^{prime} ), (c) the…

Explanation:

Step1: Differentiate using the quotient rule

The quotient rule states that if (y = \frac{u}{v}), then (y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}). Here, (u = 1), (u^\prime=0), (v=x + 8), (v^\prime = 1). [ \begin{align*} f^\prime(x)&=\frac{0\times(x + 8)-1\times1}{(x + 8)^{2}}\ &=\frac{-1}{(x + 8)^{2}} \end{align*} ]

Step2: Find partition numbers

Partition numbers of (f^\prime(x)) are the values of (x) where (f^\prime(x)) is either (0) or undefined. Set (f^\prime(x)=0): (\frac{-1}{(x + 8)^{2}}=0), no solution since (-1\neq0). (f^\prime(x)) is undefined when (x+8 = 0), i.e., (x=-8).

Step3: Find critical numbers

Critical numbers of (f) are the values of (x) in the domain of (f) where (f^\prime(x)=0) or (f^\prime(x)) is undefined. The domain of (f(x)=\frac{1}{x + 8}) is (x\neq-8). Since (f^\prime(x)) is never (0) and (x=-8) is not in the domain of (f), there are no critical numbers.

Answer:

(A) (f^\prime(x)=\frac{-1}{(x + 8)^{2}}) (B) Partition number: (x=-8) (C) No critical numbers.