suppose that ( f(x) ) and ( g(x) ) are differentiable functions such that ( f(7)=1, f^{prime}(7)=6, g(7)=5…

suppose that ( f(x) ) and ( g(x) ) are differentiable functions such that ( f(7)=1, f^{prime}(7)=6, g(7)=5 ), and ( g^{prime}(7)=3 ). find ( h^{prime}(7) ) when ( h(x)=\frac{f(x)}{g(x)} ).\n( h^{prime}(7)=square ) (simplify your answer.)

suppose that ( f(x) ) and ( g(x) ) are differentiable functions such that ( f(7)=1, f^{prime}(7)=6, g(7)=5 ), and ( g^{prime}(7)=3 ). find ( h^{prime}(7) ) when ( h(x)=\frac{f(x)}{g(x)} ).\n( h^{prime}(7)=square ) (simplify your answer.)

Answer

Explanation:

Step1: Apply the quotient rule

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

Step2: Substitute (x = 7)

We know that (f(7) = 1), (f^{\prime}(7)=6), (g(7)=5), and (g^{\prime}(7)=3). Substitute these values into the quotient - rule formula: (h^{\prime}(7)=\frac{f^{\prime}(7)g(7)-f(7)g^{\prime}(7)}{[g(7)]^{2}}) (=\frac{6\times5 - 1\times3}{5^{2}})

Step3: Simplify the expression

First, calculate the numerator: (6\times5-1\times3=30 - 3=27). The denominator is (5^{2}=25).

Answer:

(\frac{27}{25})