Calculus and Its Applications, Chapter 1, 1.8, Section 1.8, Problem 14

Determine the $f'(x)$ of the function $\displaystyle f(x) = x^3 - \frac{5}{x}$
We have $f(x) = x^3 - 5x^{-1}$, so

$\begin{equation} \begin{aligned} f'(x) = \frac{d}{dx} \left( x^3 - 5x^{-1} \right) &= \frac{d}{dx} (x^3) - 5 \cdot \frac{d}{dx} (x^{-1})\\ \\ &= 3x^2 - 5(-1) x^{-1-1} \\ \\ &= 3x^2 + 5x^{-2} \text{ or } 3x^2 + \frac{5}{x^2} \end{aligned} \end{equation}$


Then,

$\begin{equation} \begin{aligned} f''(x) &= 3 \cdot \frac{d}{dx} (x^2) + 5 \cdot \frac{d}{dx} (x^{-2}) \\ \\ &= 3 \cdot 2 x^{2-1} + 5(-2) x^{-2-1} \\ \\ &= 6x - 10 x^{-3} \text{ or } 6x - \frac{10}{x^3} \end{aligned} \end{equation}$