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

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



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