When y isn’t isolated (e.g. x² + y² = 25), differentiate both sides with respect to x, treating y as a function of x, every y term picks up a dy/dx (chain rule). Worked example Circle Find dy/dx for x² + y² = 25. Differentiate term by term: 2x + 2y·(dy/dx) = 0. Solve: 2y (dy/dx) = −2x. dy/dx = −x/y. ✓ dy/dx = −x/y (the tangent to a circle is perpendicular to the radius). ✕
Forgetting the dy/dx factor on a y-term
d/dx(y²) is 2y·(dy/dx), not just 2y — y is an (unknown) function of x, so differentiating it with respect to x always needs the chain rule. Dropping that factor is the single most common slip in this section, and it silently turns a correct method into a wrong answer. Worked example A mixed term: product rule AND implicit Find dy/dx for xy + y² = 7. Differentiate xy with the product rule: d/dx(xy) = 1·y + x·(dy/dx) = y + x(dy/dx). Differentiate y² with the chain rule: d/dx(y²) = 2y·(dy/dx). So: y + x(dy/dx) + 2y(dy/dx) = 0. Collect the dy/dx terms: (x + 2y)(dy/dx) = −y. ✓ dy/dx = −y / (x + 2y).