Single Variable Calculus, Chapter 3, 3.3, Section 3.3, Problem 25

Differentiate $\displaystyle F(y) = \left( \frac{1}{y^2} - \frac{3}{y^4} \right)(y + 5y^3)$



$
\begin{equation}
\begin{aligned}

F(y) =& \left( \frac{1}{y^2} - \frac{3}{y^4} \right)(y + 5y^3)
&& \text{Expand the equation}
\\
\\
F(y) =& \left( \frac{y}{y^2} - \frac{3y}{y^4} + \frac{5y^3}{y^2} - \frac{15y^3}{y^4} \right)
&& \text{Reduce to lowest term}
\\
\\
F(y) =& \frac{1}{y} - \frac{3}{y^3} + 5y - \frac{15}{y} = y^{-1} - 3y^{-3} + 5y - 15y^{-1}
&& \text{Combine like terms}
\\
\\
F(y) =& -3y^{-3} + 5y - 14y^{-1}
&& \text{Apply Power Rule}
\\
\\
F'(y) =& -3 \frac{d}{dy} (y^{-3}) + 5 \frac{d}{dy} (y) - 14 \frac{d}{dy} (y^{-1})
&&
\\
\\
F'(y) =& (-3)(-3y^{-4}) + (5)(1) - (14)(-y^{-2})
&& \text{Simplify the equation}
\\
\\
F'(y) =& 9y^{-4} + 5 + 14 y^{-2}
&&



\end{aligned}
\end{equation}
$