Many important curves in mathematics cannot be written as y = f(x). Instead, they are defined implicitly by equations like x2 + y2 = r2 (a circle).
The remarkable thing is that tangent lines still exist at every smooth point on these curves! We use implicit differentiation to find dy/dx without ever solving for y.
Try it: Drag the point around different curves. Notice how the tangent line and dy/dx value update continuously, even when the point is on a part of the curve where y is not a function of x.