Owl: An oval running track isn't the graph of a single function y = f(x) — for many x there are two y's. Yet at a point on the railing, the path still has a tangent direction.
Implicit differentiation lets us find dy/dx from an equation F(x,y)=0 without first solving for y. We differentiate both sides, treating y as a quiet function of x, so every y carries a y' by the chain rule.
The classic fear: "I must solve for y first." Often you can't nicely — and you don't need to.
Look at x² + 4y² = 36, an ellipse. Can that curve have a slope at (0,3) even though it fails the vertical-line test globally?