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When deriving the formula for the derivative of a parametric curve, in the form of $x = x(t)$ and $y = y(t)$, the chain rule is applied to $\frac{dy}{dt}$ to obtain $\frac{dy}{dt} = \frac{dy}{dx}\cdot\frac{dx}{dt}$, from which the slope of $y$ with respect to $x$ can be obtained.

My question is: why is it legal to use $\frac{dy}{dx}$ when $y$ is often not a function of $x$?

For example, the curve described by the parametric equations $x=6sin(t)$ and $y=t^2+t$ (image here) is clearly not a function of $x$, since it fails the vertical line test infinitely many times, and yet $\frac{dy}{dx} = \frac{2t+1}{6cos(t)}$.

What is the intuition behind $\frac{dy}{dx}$ in this case? I usually think of $\frac{dy}{dx}$ as the (unique) slope induced from a small change in $x$, but that doesn't make sense here, since a small change in $x$ corresponds to infinitely many changes in $y$.

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    Its because the small change in $x$ will change the value of $t$ which will in turn affect that value of $y$. – 2017-01-19

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The important facts are that both $x$ and $y$ are functions of $t$ and for each value of $t$ there is only one point of the curve. If the curve has as non-horizontal tangent at that point, then the slope of the tangent will be found by substituting the value of $t$ into the equation for $\dfrac{dy}{dx}$ at that point.

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    This helps a lot. I'm still having a bit of trouble interpreting the left side of the equation $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$. The right side makes complete sense to me. Obviously they're equivalent, but an equality is only useful to the extent that it provides differing perspectives of the same thing. I suppose $\frac{dy}{dx}$ can be thought of as simply a symbol to describe the slope of a graph produced using $t$, since $y$ and $x$ alone don't have any deeper meaning outside of their dependence on $t$. – 2017-01-19
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    @user407691 That is not a bad way of thinking about it. – 2017-01-19
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There is an important theorem in calculus which says, basically, that for any point on a (nice) curve where the curve is non-vertical and not self-intersecting, there is a piece of the curve surrounding that point where the curve looks like the graph of a function. In other words, there is a function $y(x)$ whose graph coincides with that small piece of the curve. Since derivatives only care about the immediate surroundings of any given point, this lets you make sense of $\frac{dy}{dx}$ at that point.

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    Thanks for the reply! This makes sense to me and seems plausible...however, if you subdivided the curve above into an infinite number of separate differentiable functions $f_1(x), f_2(x) ... $, I'm having trouble understanding what $\frac{dy}{dx}$ would represent for a particular $\Delta x = x_i - x_{i-1}$. Which function would $\Delta x$ refer to in this case? – 2017-01-19