Question

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)} ).
( h^{prime}(7)=square ) (simplify your 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:

\(\frac{27}{25}\)