Calculus and Its Applications, Chapter 1, 1.6, Section 1.6, Problem 36

Differentiate $\displaystyle G(t) = (3t^5 - t^2) \left( t - \frac{5}{t} \right)$

$\begin{equation} \begin{aligned} G(t) &= \left( 3t^5 \right) (t) - \left( 3t^5 \right) \left( \frac{5}{t} \right) - \left( t^2 \right) (t) + \left( t^2 \right) \left( \frac{5}{t} \right) && \text{Use FOIL method}\\ \\ G(t) &= 3t^6 - 15t^4 - t^3 + 5t && \text{Multiply variables with same bases by adding their exponents and divide variables with the same base by subtracting the exponents} \end{aligned} \end{equation}$


Thus, by taking the derivative, we get

$\begin{equation} \begin{aligned} G'(t) &= \frac{d}{dt} \left[ 3t^6 - 15t^4 - t^3 + 5t \right]\\ \\ &= 18t^5 - 60 t^3 - 3t^2 + 5 \end{aligned} \end{equation}$