find the derivative of the function z, below. it may be to your advantage to simplify first. $z = \\frac{2t…

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $z=\frac{u}{v}$, then $\frac{dz}{dt}=\frac{u'v - uv'}{v^{2}}$, where $u = 2t + 14$, $u'=2$, $v = 7t+3$, and $v' = 7$.

Step2: Substitute values into quotient - rule formula

$\frac{dz}{dt}=\frac{2(7t + 3)-(2t + 14)\times7}{(7t + 3)^{2}}$.

Step3: Expand the numerator

First, expand $2(7t + 3)=14t+6$ and $(2t + 14)\times7 = 14t+98$. Then $\frac{dz}{dt}=\frac{14t + 6-(14t + 98)}{(7t + 3)^{2}}$.

Step4: Simplify the numerator

$14t + 6-(14t + 98)=14t+6 - 14t-98=-92$. So, $\frac{dz}{dt}=\frac{-92}{(7t + 3)^{2}}$.

Answer:

$\frac{-92}{(7t + 3)^{2}}$