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Imagine tracking a roller coaster looping through space or a planet tracing an elliptical orbit. You cannot write these paths as a simple function y=f(x)y = f(x) because a single xx-value often has multiple yy-values.
Instead, curves in nature are either implicitly definedβ€”where xx and yy are bound together in an equationβ€”or parametrically defined, where both coordinates unfold over time tt.
How do you pinpoint the exact line of momentum (the tangent) or the perpendicular line of force (the normal) along these complex curves?

The Geometry of Tangents and Normals

A tangent line grazes a curve at a single point (x0,y0)(x_0, y_0), matching the curve's instantaneous steepness m=dydxm = \frac{dy}{dx} at that exact location.
The normal line is perpendicular to the tangent at that same point, meaning its slope is the negative reciprocal mβŠ₯=βˆ’1mm_{\perp} = -\frac{1}{m}.
Once you know any point (x0,y0)(x_0, y_0) and its slope mm, the point-slope formula gives the line: yβˆ’y0=m(xβˆ’x0)y - y_0 = m(x - x_0)
For the normal line, you simply swap the slope mm for βˆ’1m-\frac{1}{m}: yβˆ’y0=βˆ’1m(xβˆ’x0)y - y_0 = -\frac{1}{m}(x - x_0)
What happens when you cannot isolate yy to take a standard derivative? Let's unlock implicit curves.

Tangents to Implicit Curves

In 1638, RenΓ© Descartes challenged Pierre de Fermat to find the tangent line to the looped curve x3+y3=3axyx^3 + y^3 = 3axy, now called the Folium of Descartes.
To find dydx\frac{dy}{dx} without solving for yy, use implicit differentiation: differentiate both sides with respect to xx, applying the chain rule ddx[yn]=nynβˆ’1dydx\frac{d}{dx}[y^n] = n y^{n-1} \frac{dy}{dx} whenever differentiating a term with yy.