Exploring Behaviors of Implicit Relations AB & BC

Implicit differentiation produces a derivative in two variables, usually a quotient, so it has two features worth reading: the numerator vanishing gives a horizontal tangent and the denominator vanishing gives a vertical one. For $x^{2} + y^{2} = 25$, $\frac{dy}{dx} = -\frac{x}{y}$, so horizontal tangents need $x = 0$ and vertical tangents need $y = 0$. Each condition is only half an answer: substituting back into the original equation turns $x = 0$ into the two points $(0, 5)$ and $(0, -5)$, and rules out $(0, 3)$, which meets the condition and is not on the curve.

A relation is not a function, so a slope belongs to a point rather than to an input, and one condition can produce several points. Concavity is still the sign of $\frac{d^{2}y}{dx^{2}}$ and never the sign of $\frac{dy}{dx}$. Computing it takes one extra move: differentiate the quotient, then substitute $\frac{dy}{dx}$ back in. For the circle that gives $-\frac{25}{y^{3}}$, so the upper semicircle is concave down and the lower one is concave up, even though both bend toward the same centre.