Question

For \(x=a\cos^3\theta\) and \(y=a\sin^3\theta\), which expression gives \(\frac{dy}{dx}\) in terms of \(x\) and \(y\)?

Options

• \(\sqrt[3]{\frac{x}{y}}\)

• \(\sqrt[3]{\frac{y}{x}}\)

• \(-\sqrt[3]{\frac{y}{x}}\)

• \(-\sqrt[3]{\frac{x}{y}}\)

Answer 1

From the parametric equations, \(\tan\theta=\sqrt[3]{\frac{y}{x}}\). Therefore, since \(\frac{dy}{dx}=-\tan\theta\), the derivative is \(-\sqrt[3]{\frac{y}{x}}\).