implicit differentiation

Some curves can't be tidily written as 'y equals some formula in x'. A circle is the classic case: x^2 + y^2 = 1 describes it perfectly, but solving for y forces you to split it into two pieces (a top half and a bottom half) with an awkward square root. Implicit differentiation is a trick that lets you find the slope on such a curve without ever untangling it into y = something.

The idea is to treat y as a function of x even though it's hidden, then differentiate both sides of the equation with respect to x, applying the chain rule every time y appears. For x^2 + y^2 = 1, differentiating gives 2x + 2y(dy/dx) = 0 — note the dy/dx that the chain rule attaches to the y^2 term. Then you solve algebraically for dy/dx, getting dy/dx = -x/y. The result usually depends on both x and y, which is fine: plug in any point on the curve to get the slope there.

The one habit that makes this work is remembering that y is secretly a function of x, so every y carries a dy/dx when you differentiate it — that factor is the whole point. A term like y^3 differentiates to 3y^2(dy/dx), not 3y^2. Implicit differentiation is also how related-rates problems get solved, and it underlies the rule for differentiating inverse functions.