Question

the following table lists the values of functions $f$ and $g$, and of their derivatives, $f$ and $g$, for $x = 4$.
let function $f$ be defined as $f(x)=\frac{f(x)}{g(x)}$.
$f(4)=\square$

Explanation:

Step1: Apply the quotient rule

The quotient rule states that if \(F(x)=\frac{f(x)}{g(x)}\), then \(F'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^{2}}\).

Step2: Substitute \(x = 4\) into the quotient - rule formula

We know that \(f(4)=-4\), \(g(4)=-2\), \(f'(4) = 0\), and \(g'(4)=8\).
Substitute these values into \(F'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^{2}}\):

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Step3: Simplify the expression

First, calculate the numerator: \(0\times(-2)-(-4)\times8=0 + 32=32\).
Then, calculate the denominator: \((-2)^{2}=4\).
So, \(F'(4)=\frac{32}{4}\)

Answer:

\(8\)