Solution — 2021 Question 2 Differentiation & Applications| Integration & Applications

(a)

\begin{matrix} y & = {(2 x^{3} - 5 x^{\frac{1}{3}})}^{2} \\ = 4 x^{6} - 20 x^{\frac{10}{3}} + 25 x^{\frac{2}{3}} \\ \frac{d y}{d x} & = 24 x^{5} - \frac{200}{3} x^{\frac{7}{3}} + \frac{50}{3} x^{- \frac{1}{3}} \\ = 24 x^{5} - \frac{200}{3} x^{\frac{7}{3}} + \frac{50}{3 x^{\frac{1}{3}}} ∎ \end{matrix}

(b)

\begin{matrix} \int \frac{1}{\sqrt{5 - 4 x}} d x \\ = \int {(5 - 4 x)}^{- \frac{1}{2}} d x \\ = - \frac{{(5 - 4 x)}^{\frac{1}{2}}}{2} + c ∎ \end{matrix}