If x and y are connected parametrically by the equations given, without eliminating the parameter, Find `(dy)/(dx)` . `x=(sin^3t)/(sqrt(cos2t)), y=(cos^3t)/(sqrt(cos2t))`

Explanation

The given equations are
`x=frac{sin ^{3} t}{sqrt{cos 2 t}}` and `y=frac{cos ^{3} t}{sqrt{cos 2 t}}`

Then,
`frac{d x}{d t}=frac{d}{d t}[frac{sin ^{3} t}{sqrt{cos 2 t}}]`

`=frac{sqrt{cos 2 t} cdot frac{d}{d t}(sin ^{3} t)-sin ^{3} t cdot frac{d}{d t} sqrt{cos 2 t}}{cos 2 t}`

`=frac{sqrt{cos 2 t} cdot 3 sin ^{2} t cdot frac{d}{d t}(sin t)-sin ^{3} t times frac{1}{2 sqrt{cos 2 t}} cdot frac{d}{d t}(cos 2 t)}{cos 2 t}`

`=frac{3 sqrt{cos 2 t} cdot sin ^{2} t cos t-frac{sin ^{3} t}{2 sqrt{cos 2 t}} cdot(-2 sin 2 t)}{cos 2 t}`

`=frac{3 cos 2 t sin ^{2} t cos t+sin ^{3} t sin 2 t}{cos 2 t sqrt{cos 2 t}}`

`frac{d y}{d t}=frac{d}{d t}[frac{cos ^{3} t}{sqrt{cos 2 t}}]`

`=frac{sqrt{cos 2 t} cdot frac{d}{d t}(cos ^{3} t)-cos ^{3} t cdot frac{d}{d t}(sqrt{cos 2 t})}{cos 2 t}`

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