Question

homework3: problem 25
(1 point)
consider the functions ( f(x) ) and ( g(x) ), for which ( f(0)=6, g(0)=3, f^{prime}(0)=5 ), and ( g^{prime}(0)=-5 ).
find ( h^{prime}(0) ) for the function ( h(x)=\frac{f(x)}{g(x)} ).
( h^{prime}(0)= )
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Explanation:

Step1: Apply quotient rule

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

Step2: Substitute \(x = 0\)

Substitute \(x = 0\) into the formula for \(h'(x)\). We know that \(f(0) = 6\), \(g(0)=3\), \(f'(0)=5\), and \(g'(0)=- 5\).

$$
LATEXBLOCK0
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Answer:

\(5\)