Question

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

( f(x)=\frac{1}{x + 8} )

(a) ( f^{prime}(x)= )

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, \(u = 1\), \(u^\prime=0\), \(v=x + 8\), \(v^\prime = 1\).

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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 \(-1
eq0\).
\(f^\prime(x)\) 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 \(x
eq-8\). 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.