homework3: problem 25\n(1 point)\nconsider the functions ( f(x) ) and ( g(x) ), for which ( f(0)=6, g(0)=3…

homework3: problem 25\n(1 point)\nconsider the functions ( f(x) ) and ( g(x) ), for which ( f(0)=6, g(0)=3, f^{prime}(0)=5 ), and ( g^{prime}(0)=-5 ).\nfind ( h^{prime}(0) ) for the function ( h(x)=\frac{f(x)}{g(x)} ).\n( h^{prime}(0)= )\npreview my answers submit answers\nyou have attempted this problem 0 times.\nyou have unlimited attempts remaining.\nemail instructor

homework3: problem 25\n(1 point)\nconsider the functions ( f(x) ) and ( g(x) ), for which ( f(0)=6, g(0)=3, f^{prime}(0)=5 ), and ( g^{prime}(0)=-5 ).\nfind ( h^{prime}(0) ) for the function ( h(x)=\frac{f(x)}{g(x)} ).\n( h^{prime}(0)= )\npreview my answers submit answers\nyou have attempted this problem 0 times.\nyou have unlimited attempts remaining.\nemail instructor

Answer

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). [ \begin{align*} h'(0)&=\frac{f'(0)g(0)-f(0)g'(0)}{[g(0)]^{2}}\ &=\frac{(5)(3)-(6)(-5)}{3^{2}}\ &=\frac{15 + 30}{9}\ &=\frac{45}{9} \end{align*} ]

Answer:

(5)