Question

find the derivative.
y = \frac{7x + 6}{6x - 19}
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, $u = 7x+6$, $v = 6x - 19$.

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

Differentiate $u = 7x+6$ with respect to $x$, $u'=\frac{d}{dx}(7x + 6)=7$. Differentiate $v = 6x - 19$ with respect to $x$, $v'=\frac{d}{dx}(6x - 19)=6$.

Step3: Apply quotient - rule

Substitute $u$, $v$, $u'$, and $v'$ into the quotient - rule formula: $y'=\frac{7(6x - 19)- (7x + 6)\times6}{(6x - 19)^2}$.

Step4: Expand the numerator

Expand $7(6x - 19)- (7x + 6)\times6$: $42x-133-(42x + 36)=42x-133 - 42x-36$.

Step5: Simplify the numerator

$42x-133 - 42x-36=-161$. So, $y'=\frac{-161}{(6x - 19)^2}$.

Answer:

$y'=\frac{-161}{(6x - 19)^2}$