${\left({\mathit{x}}^2+{\mathit{y}}^2\right)}^2$ = xy
differentiating w.r.t. x
⇒ 2 $\left({\mathit{x}}^2+{\mathit{y}}^2\right)$ $\left(2\mathit{x}+2\mathit{y}\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}\right)$ = y + x $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$
⇒ $4{\mathit{x}}^3+4{\mathit{x}}^2\mathit{y}\mathit{y}\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ + $4{\mathit{y}}^2\mathit{x}$ + $4{\mathit{y}}^3\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ - y - x $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = 0
⇒ $\left(4{\mathit{x}}^2\mathit{y}+4{\mathit{y}}^3-\mathit{x}\right)$ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ + $4{\mathit{x}}^3+4{\mathit{y}}^2\mathit{x}$ - y = 0
⇒ $\left(4{\mathit{x}}^2\mathit{y}+4{\mathit{y}}^3-\mathit{x}\right)$ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = - $4{\mathit{x}}^3-4{\mathit{y}}^2\mathit{x}$ + y
⇒ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{-4{\mathit{x}}^3-4{\mathit{y}}^2\mathit{x}+\mathit{y}}{4{\mathit{x}}^2\mathit{y}+4{\mathit{y}}^3-\mathit{x}}$
OR
x = a (2θ - sin 2θ)
y = a (1 - cos θ)
differentiating w.r.t. θ
$\frac{\mathit{d}\mathit{x}}{\mathit{d}\mathit{\theta }}$ = a (2 - cos 2θ × 2)
$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{\theta }}$ = a (+ sin 2θ × 2)
$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{\left(+\mathit{s}\mathit{i}\mathit{n}2\mathit{\theta }\times 2\right)}{\left(-2\mathit{c}\mathit{o}\mathit{s}2\mathit{\theta }\times 2\right)}$ = $\frac{\left(\mathit{s}\mathit{i}\mathit{n}2\mathit{\theta }\right)}{\left(1-\mathit{c}\mathit{o}\mathit{s}2\mathit{\theta }\right)}$
$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{2\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\times \mathit{c}\mathit{o}\mathit{s}\mathit{\theta }}{2\mathit{s}\mathit{i}{\mathit{n}}^2\mathit{\theta }}$
$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }}{\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }}$ = cot θ
at , θ = $\frac{\mathit{\pi }}{3}$
$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = cot $\frac{\mathit{\pi }}{3}$ = $\frac{1}{\sqrt{3}}$