Implicit Differentiation

x² + y² = 25 is a circle. It fails the vertical-line test — it's not a function of x. You can't cleanly solve for y (it's ±). And yet there's a perfectly good slope at every point on the circle. How do you find dy/dx when y is stuck INSIDE the equation? You differentiate both sides as-is, treating y as a function of x, and round up the dy/dx terms. That's implicit differentiation.