suppose that f(5) = 1, f (5) = 7, g(5) = -9, and g(5) = 5. find the following values. (a) (fg)(5) (b)…

Explanation:

Step1: Recall product - rule

The product - rule states that $(fg)'(x)=f'(x)g(x)+f(x)g'(x)$. Substitute $x = 5$, $f(5)=1$, $f'(5)=7$, $g(5)=-9$, and $g'(5)=5$ into the formula: $(fg)'(5)=f'(5)g(5)+f(5)g'(5)$ $=7\times(-9)+1\times5$ $=-63 + 5$ $=-58$

Step2: Recall quotient - rule

The quotient - rule states that $(\frac{f}{g})'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{g^{2}(x)}$. Substitute $x = 5$ into the formula: $(\frac{f}{g})'(5)=\frac{f'(5)g(5)-f(5)g'(5)}{g^{2}(5)}$ $=\frac{7\times(-9)-1\times5}{(-9)^{2}}$ $=\frac{-63 - 5}{81}$ $=\frac{-68}{81}$

Step3: Recall quotient - rule for $\frac{g}{f}$

The quotient - rule states that $(\frac{g}{f})'(x)=\frac{g'(x)f(x)-g(x)f'(x)}{f^{2}(x)}$. Substitute $x = 5$ into the formula: $(\frac{g}{f})'(5)=\frac{g'(5)f(5)-g(5)f'(5)}{f^{2}(5)}$ $=\frac{5\times1-(-9)\times7}{1^{2}}$ $=\frac{5 + 63}{1}$ $=68$

Answer:

(a) - 58 (b) $-\frac{68}{81}$ (c) 68