Single Variable Calculus, Chapter 3, 3.3, Section 3.3, Problem 24

Determine $Y(u) = (u^{-2} + u^{-3})(u^5 - 2u^2)$


$
\begin{equation}
\begin{aligned}

y(u) =& (u^{-2} + u^{-3})(u^5 - 2u^2)
&& \text{Expand the equation}
\\
\\
y(u) =& (u^{-2 + 5} + u^{-3 + 5} - 2u^{-2 + 2} - 2u^{-3 + 2})
&& \text{Simplify the equation}
\\
\\
y(u) =& u^3 + u^2 - 2u^0 - 2u^{-1}
&& \text{Apply Power Rule}
\\
\\
y'(u) =& \frac{d}{du} (u^3) + \frac{d}{du} (u^2) - 2 \frac{d}{du} (u^-1) - \frac{d}{du} (2)
&&
\\
\\
y'(u) =& 3u^2 + 2u - (2)(-1)(u^{-2}) - 0
&& \text{Simplify the equation}
\\
\\
y'(u) =& 3u^2 + 2u + 2u^{-2}

\end{aligned}
\end{equation}
$