Question

Suppose that f(5) = 1, f '(5) = 4, g(5) = −9, and g'(5) = 5. Find the following values.

Answer 1

$\bf \begin{cases} f(5)=1\qquad &f'(5)=4 \\\\ g(5)=-9\qquad &g'(x)=5 \end{cases} \\\\ \begin{cases} (f\cdot g)'(5)\impliedby \textit{product rule} \\\\ \left( \frac{f}{g} \right)'(5)\impliedby \textit{quotient rule} \end{cases}\\\\ -----------------------------\\\\ (f\cdot g)'(5)\implies f(x)\cdot g(x)\implies \begin{array}{llll} f'(x)g(x)+f(x)g'(x)\\\\ x=5\\\\\ [4\cdot -9]+[1\cdot 5] \end{array} \\\\ -----------------------------$

$\bf \left( \frac{f}{g} \right)'(5)\implies \begin{array}{llll} \cfrac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}\\\\ x=5\\\\ \cfrac{[4\cdot -9]-[1\cdot 5]}{(-9)^2} \end{array}$