Question

the table above gives values for the functions f and g and their derivatives at x = 3. let k be the function given by k(x) = \frac{f(x)}{g(x)}, where g(x) ≠ 0. what is the value of k(3)?

Explanation:

Step1: Recall the quotient rule

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

Step2: Substitute \(x = 3\) into the quotient - rule formula

We know from the table that \(f(3)=-1\), \(g(3)=2\), \(f^{\prime}(3)=5\), and \(g^{\prime}(3)=-2\).
Substitute these values into the formula \(k^{\prime}(3)=\frac{f^{\prime}(3)g(3)-f(3)g^{\prime}(3)}{[g(3)]^{2}}\).

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Answer:

\(2\)