Implicit Differentiation

A circle like x² + y² = 25 is not a function, since one x-value can give two y-values. But at any single point on it, the curve has one clear tangent line, one slope. Implicit differentiation finds that slope without ever solving for y. The trick is to treat y as some unnamed function of x, even though you don't know its formula, and differentiate the whole equation as it stands, term by term, with respect to x.

Because y secretly depends on x, every time you differentiate a y-term you must apply the chain rule, which tacks on a factor of dy/dx. So d/dx(y²) is 2y times dy/dx, not just 2y. Terms with only x differentiate normally, and constants vanish as always. For x² + y² = 25, differentiating gives 2x + 2y dy/dx = 0. Solve for dy/dx by isolating it: dy/dx = -x/y. Notice the answer depends on both x and y, which makes sense, since the slope at the top of the circle differs from the slope at the side.

When the equation mixes x and y in a product, like x³ + y³ = 6xy, the right side needs the product rule before you can collect dy/dx terms. This topic is the main tool behind the rest of the unit: inverse function derivatives and related rates both lean on this same habit of differentiating an equation as written and solving for the derivative you want afterward.