Calculus and Its Applications, Chapter 1, 1.5, Section 1.5, Problem 48

If $\displaystyle y = x^{-\frac{3}{4}} - 3x^{\frac{2}{3}} + x^{\frac{5}{4}} + \frac{2}{x^4}$, find $y'$

$\begin{equation} \begin{aligned} y' &= \frac{d}{dx} \left( x^{-\frac{3}{4}} - 3x^{\frac{2}{3}} + x^{\frac{5}{4}} + \frac{2}{x^4} \right)\\ \\ &= \frac{d}{dx} \left( x^{-\frac{3}{4}} \right) - \frac{d}{dx} \left( 3x^{\frac{2}{3}}\right) + \frac{d}{dx} \left( x^{\frac{5}{4}} \right)+ \frac{d}{dx} \left( \frac{2}{x^4} \right)\\ \\ &= \frac{d}{dx} \left( x^{-\frac{3}{4}} \right) - 3 \cdot \frac{d}{dx} \left( x^{\frac{2}{3}} \right) + \frac{d}{dx} \left( x^{\frac{5}{4}} \right) + 2 \cdot \frac{d}{dx} \left( x^{-4} \right)\\ \\ &= -\frac{3}{4} \cdot x^{-\frac{3}{4}-1} - 3 \cdot \frac{2}{3} x^{\frac{2}{3} - 1} + \frac{5}{4} \cdot x^{\frac{5}{4}-1} + 2 \cdot (-4) x^{-4-1}\\ \\ &= -\frac{3}{4}x^{-\frac{7}{4}} - 2 x^{-\frac{1}{3}} + \frac{5}{4} x^{\frac{1}{4}} - 8x^{-5}\\ \\ &= \frac{-3}{4x^{\frac{7}{4}}} - \frac{2}{x^{\frac{1}{3}}} + \frac{5}{4} x^{\frac{1}{4}} - \frac{8}{x^5} \end{aligned} \end{equation}$