Tangent Lines to Implicit Curves

An implicit curve like x² + xy + y² = 7 is not a function, so it can loop back on itself and have tangent lines pointing in every direction, including straight up. A regular function y = f(x) can never have a vertical tangent, because that would mean two y-values for one x-value. But an implicit curve is allowed to do that, which is exactly why it can have a point where the tangent is vertical: the curve momentarily behaves like it has infinite slope, which is the signature of a function failing to be a function at that spot.

The method is always the same. Differentiate both sides of the equation with respect to x, treating y as a hidden function of x and using the chain rule every time you touch a y, then solve the resulting equation for dy/dx. The result is almost always a fraction built from x and y. That fraction tells you everything: plug in a specific point to get the slope there for a tangent line equation, set the numerator equal to 0 to find horizontal tangents (slope zero), and set the denominator equal to 0 to find vertical tangents (slope undefined). In every case you still need to check that the point actually lies on the original curve.

This topic is the payoff for learning implicit differentiation in the previous lesson: now that you can compute dy/dx without solving for y, you can do all the same tangent-line and shape analysis you did for ordinary functions. It sets up later work on related rates and curve analysis, where implicit relationships between changing quantities are the norm rather than the exception.