Suppose [math]y[/math] is a differentiable function of [math]x[/math] and [math]x[/math] is a differentiable function of [math]t[/math]. Apply the chain rule to write [math]y'\left(t\right)[/math] in terms of [math]y'\left(x\right)[/math] and [math]x'\left(t\right)[/math].
[math]y'\left(t\right)=y'\left(x\right)x'\left(t\right)[/math][br][br]We can rearrange this to read:[br][math]y'\left(x\right)=\frac{y'\left(t\right)}{x'\left(t\right)}[/math][br][br]The takeaway here is that if the graph of some function [math]y=f\left(x\right)[/math] can be parameterized by a differentiable path [math]\vec{c}\left(t\right)=\left(x\left(t\right),y\left(t\right)\right)[/math], then the derivative of [math]f[/math] at any point along the graph is the slope of the velocity vector to the curve at that point!
There are a couple pretty big advantages to this observation. For one, many parameterized curves will pass the vertical line test, but finding a rectangular equation describing [math]y[/math] as a function of [math]x[/math] is a difficult task. The cycloid curve is a great example. We are now able to find the tangent line to such a curve without having to execute the tedious algebra required to write [math]y[/math] as a function of [math]x[/math]. [br][br]Moreover, there are many parameterized curves that fail the vertical line test. However if there is a window about a point on the curve for which [math]y[/math] can be written as a function of [math]x[/math], we now have a way of defining and understanding a tangent line and slope.[br][br]In the GeoGebra applet below you can enter a parameterized curve, adjust the domain via a slider, and enter what you believe to be the slope of the tangent line. The animation will display a point moving along the curve, together with the velocity vector and a proposed tangent line. If you've entered everything in correctly the tangent line should appear to be parallel to the velocity vector. You can use the first 20 problems from Section 11.10.1 in [url=https://www.stitz-zeager.com/szprecalculus07042013.pdf]Stitz-Zeager[/url] to practice finding tangent line equations.