C CM_Tutor Moderator Moderator Joined Mar 11, 2004 Messages 2,642 Gender Male HSC N/A Oct 16, 2021 #2
} \\ \\ y &= x^3 - 1 \\ y + 1 &= x^3 \\ x &= \sqrt[3]{y + 1} \\ \cfrac{dx}{dy} &= \cfrac{1}{3}(y + 1)^{-\frac{2}{3}} \times 1 \\ &= \cfrac{1}{3\sqrt[3]{(y + 1)^2}} \qquad \text{. . . . . . . . . . (2)} \\ \\ \cfrac{dy}{dx} \times \cfrac{dx}{dy} &= 3x^2 \times \cfrac{1}{3\sqrt[3]{(y + 1)^2}} \qquad \text{using (1) and (2)} \\ &= \cfrac{x^2}{\sqrt[3]{\left(x^3\right)^2}} \qquad \text{noting $y + 1 = x^3$} \\ &= \cfrac{x^2}{\sqrt[3]{\left(x^2\right)^3}} \\ &= \cfrac{x^2}{x^2} \\ &= 1 \qquad \text{as required} \end{align*})
