Question

If $x=a{\sin }^3\theta$ and $y=a{\cos }^3\theta$, then find the value of $\frac{dy}{dx}$.

Answer 1

Given $x=a{\sin }^3\theta$

Differentiating w.r.t. $\theta$ we get

$\frac{dx}{d\theta }=3a{\sin }^2\theta \left(\cos \theta \right)$

$\Rightarrow \frac{dx}{d\theta }=3a{\sin }^2\theta \cos \theta$

Again

Differentiating w.r.t. $\theta$ we get

$\frac{dy}{d\theta }=3a{\cos }^2\theta (-\sin \theta )$

$\Rightarrow \frac{dy}{d\theta }=-3a{\cos }^2\theta \sin \theta$

$\therefore \frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }$

$=-\frac{3a{\cos }^2\theta \sin \theta }{3a{\sin }^2\theta \cos \theta }$

$=-\frac{-\cos \theta }{\sin \theta }$

$\therefore \frac{dy}{dx}=-\cot \theta$.