Single Variable Calculus, Chapter 3, 3.3, Section 3.3, Problem 40

Differentiate $\displaystyle y = \frac{u^6 - 2u^3 + 5}{u^2}$



$\begin{equation} \begin{aligned} y' =& \frac{u^2 \displaystyle \frac{d}{du} (u^6 - 2u^3 + 5) - \left[ (u^6 - 2u^3 + 5) \frac{d}{du} (u^2) \right]}{(u^2)^2} && \text{Apply Quotient Rule} \\ \\ y' =& \frac{(u^2)(6u^5 - 6u^2) - [(u^6 - 2u^3 + 5)(2u)]}{u^4} && \text{Expand the equation} \\ \\ y' =& \frac{6u^7 - 6u^4 - 2u^7 + 4u^4 - 10u}{u^4} && \text{Combine like terms} \\ \\ y' =& \frac{4u^7 - 2u^4 - 10u}{u^4} && \text{Factor the numerator and denominator} \\ \\ y' =& \frac{\cancel{u} (4u^6 - 2u^3 - 10)}{(u^3)\cancel{(u)}} && \text{Cancel out like terms} \\ \\ y' =& \frac{4u^6 - 2u^3 - 10}{u^3} && \end{aligned} \end{equation}$