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{5}{x + 8} )

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

Explanation:

Step1: Apply the quotient rule

The quotient rule states that if \(y = \frac{u}{v}\), then \(y'=\frac{u'v - uv'}{v^{2}}\). Here, \(u = 5\), \(u'=0\), \(v=x + 8\), \(v'=1\).

$$
LATEXBLOCK0
$$

Step2: Simplify the expression

$$ f'(x)=\frac{- 5}{(x + 8)^{2}} $$

Answer:

\(f'(x)=-\frac{5}{(x + 8)^{2}}\)