find the derivative of the function. y = (7x - 9)/(3x + 1) the solution is y =

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = 7x-9$ and $v = 3x + 1$.

Step2: Find $u'$ and $v'$

Differentiate $u$ with respect to $x$: $u'=\frac{d}{dx}(7x - 9)=7$. Differentiate $v$ with respect to $x$: $v'=\frac{d}{dx}(3x + 1)=3$.

Step3: Apply quotient - rule

Substitute $u$, $u'$, $v$, and $v'$ into the quotient - rule formula. $y'=\frac{7(3x + 1)-(7x - 9)\times3}{(3x + 1)^{2}}$.

Step4: Expand and simplify

Expand the numerator: $7(3x + 1)-(7x - 9)\times3=21x+7-(21x-27)$. Then $21x + 7-21x + 27=34$. So $y'=\frac{34}{(3x + 1)^{2}}$.

Answer:

$\frac{34}{(3x + 1)^{2}}$