Single-variable calculus from first principles
Sometimes y isn't handed to you as a tidy y = f(x). Instead it's tangled into an equation, like a circle x² + y² = 25. You can still find the slope dy/dx without untangling, using implicit differentiation.
The whole move rests on one assumption: treat y as a (hidden) function of x. Then differentiate both sides of the equation with respect to x. Every time you differentiate a y-term, the chain rule tacks on a dy/dx factor, because y depends on x.
Picture a ladder leaning against a wall and starting to slide. As the foot slides out, the top slides down: the horizontal position x and the vertical position y change together, locked by the ladder's fixed length. You never solve for one in terms of the other, yet you can still relate their rates. Implicit differentiation does exactly that, differentiating an equation that ties x and y together without ever untangling y by itself.