Calculus and Its Applications, Chapter 1, 1.6, Section 1.6, Problem 28

Differentiate $\displaystyle f(t) = \frac{t}{5 + 2t} - 2t^4$

$
\begin{equation}
\begin{aligned}
f'(t) &= \frac{d}{dt} \left( \frac{t}{5 + 2t} \right) - \frac{d}{dt} (2t^4) \\
\\
&= \frac{(5 + 2t) \cdot \frac{d}{dt} (t) - (t) \cdot \frac{d}{dt} (5 + 2t)}{(5 + 2t)^2} - 8t^3\\
\\
&= \frac{(5 + 2t)(1) - (t) (2)}{(5 + 2t)^2} - 8t^3\\
\\
&= \frac{5 + 2t - 2t}{(t + 2t)^2} - 8t^3\\
\\
&= \frac{5}{(5 + 2t)^2} - 8t^3\\
\\
&= \frac{5 - 8t^3 (t + 2t)^2}{(5 + 2t)^2} \\
\\
&= \frac{5 - 8t^3 \left[ 5^2 + 2(5)(2t)+(2t)^2 \right]}{(5 + 2t)^2}\\
\\
&= \frac{5 - 8t^3 \left[ 25 + 20t + 4t^2 \right]}{(t + 2t)^2}\\
\\
&= \frac{5 - 200t^3 - 160 t^4 - 32t^5}{(5 + 2t)^2}
\end{aligned}
\end{equation}
$