Question

section 2.5: product and quotient rules (homework score: 10/80 answered: 1/8 question 2 if $f(x)=\frac{7x + 4}{5x + 2}$, find: $f(x)=$ $f(5)=$ question help: video

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. For $f(x)=\frac{7x + 4}{5x+2}$, let $u = 7x + 4$ and $v=5x + 2$. Then $u'=7$ and $v' = 5$.

Step2: Apply quotient - rule

$f'(x)=\frac{(7)(5x + 2)-(7x + 4)(5)}{(5x + 2)^{2}}$. Expand the numerator: $(7)(5x + 2)=35x+14$ and $(7x + 4)(5)=35x+20$. So $f'(x)=\frac{35x + 14-(35x + 20)}{(5x + 2)^{2}}=\frac{35x+14 - 35x-20}{(5x + 2)^{2}}=\frac{-6}{(5x + 2)^{2}}$.

Step3: Find $f'(5)$

Substitute $x = 5$ into $f'(x)$. $f'(5)=\frac{-6}{(5\times5 + 2)^{2}}=\frac{-6}{(25 + 2)^{2}}=\frac{-6}{27^{2}}=\frac{-6}{729}=-\frac{2}{243}$.

Answer:

$f'(x)=\frac{-6}{(5x + 2)^{2}}$
$f'(5)=-\frac{2}{243}$