differentiate the function.\ng(x) = \\frac{9 + 6x}{5 - 8x}\ng(x) =

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $g(x)=\frac{u(x)}{v(x)}$, then $g'(x)=\frac{u'(x)v(x)-u(x)v'(x)}{v(x)^2}$. Here, $u(x)=9 + 6x$, $v(x)=5-8x$.

Step2: Find $u'(x)$ and $v'(x)$

Differentiate $u(x)$ with respect to $x$: $u'(x)=\frac{d}{dx}(9 + 6x)=6$. Differentiate $v(x)$ with respect to $x$: $v'(x)=\frac{d}{dx}(5-8x)=-8$.

Step3: Apply quotient - rule

Substitute $u(x)$, $v(x)$, $u'(x)$, and $v'(x)$ into the quotient - rule formula. $g'(x)=\frac{6(5 - 8x)-(9 + 6x)(-8)}{(5 - 8x)^2}$.

Step4: Expand the numerator

Expand $6(5 - 8x)-(9 + 6x)(-8)$: $6(5 - 8x)=30-48x$, and $(9 + 6x)(-8)=-72-48x$. So, $g'(x)=\frac{30-48x + 72 + 48x}{(5 - 8x)^2}$.

Step5: Simplify the numerator

Combine like - terms in the numerator: $30+72-48x + 48x = 102$. So, $g'(x)=\frac{102}{(5 - 8x)^2}$.

Answer:

$\frac{102}{(5 - 8x)^2}$