Let $f$ and $g$ be the functions defined by $f(t) = 2t^{2}$ and $g(t) = 2t^{3}+8t$.

Determine $f'(t)$ and $g'(t)$.
$f'(t) =$
$g'(t) =$

Let $q(t) = \displaystyle{\frac{2t^{3}+8t}{2t^{2}}}$ and observe that $q(t) = \displaystyle{\frac{g(t)}{f(t)}}$. Rewrite the formula for $q$ by dividing each term in the numerator by the denominator and simplify to write $q(t)$ as a sum of constant multiples of powers of $t$. Then, compute $q'(t)$ using the sum and constant multiple rules.
$q'(t) =$

True or False: $q'(t) = \displaystyle{\frac{g'(t)}{f'(t)}}$ ? True False