A Slope Built From Two Derivatives

You'll learnto build the slope of a parametric curve as a quotient with the right derivative on top, to read a horizontal tangent off the numerator and a vertical tangent off the denominator, and to see what a parametrisation carries that the set of points alone does not.

When a parameter supplies both coordinates, the slope stops being one derivative and becomes a ratio of two. Which one goes on top is fixed by the chain rule rather than by taste, and the inverted version is exactly as easy to remember. The parameter also carries something the curve alone cannot: which way it is traced, and how many times each point is visited.

8 STEPS · 6 QUICK CHECKS · THE SLOPE IS A QUOTIENT, NOT A DERIVATIVE · v1

dy/dt on top · dx/dt underneath · the parameter carries the direction

Before you start

What you're looking at

One plane, drawn at equal scale on both axes. A particle traces a looped curve as the parameter t runs from −2 to 2; a ring marks where it started and a filled dot is where it is now.

The question

The curve has a slope at nearly every point, but neither coordinate is a function of the other here. Where does that slope come from, and what happens where it runs out?

Watch for

Two arrows leave the particle — one flat, one upright. Watch the upright one vanish twice and the flat one vanish once. Those three moments are the horizontal and vertical tangents, and they are read off different parts of one fraction.

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