# Explain parameter elimination for complex curves

I'm reading Stewart's Calculus and in "Calculus with Parametric Curves" he shows the formula which comes from the Chain rule:
$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$
But what I don't understand is that for some complex curves for example:
$x = 2\sin\left(1+3t\right), y = 2t^{3}$

If I try to retreive y(x) I will get
$y\ =\ 2\left(\frac{\arcsin\left(\frac{x}{2}\right)-1}{3}\right)^{3}$
Which will represent only small part of the parametric curve(graph).
And only for some small range of t  $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$. For example when t = 1, $y(x)$ is not defined.
So the question is: Am I missing something, or $\frac{dy}{dx}$ can't always represent $\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$?

• I will tip you additional 5$. Just give me some better explanation please. • I see you accepted. Do you still need aditional help? I explained above that dy/dx always equals (dy/dt)/(dx/dt). Not sure what you mean by when t = 1, y(x) is not defined. • I'm preparing another question, will be ready in couple minutes. The answer is accepted. Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer. • answered • 424 views •$5.00