Find dy/dx if sin(x + y) = xy.

1. 1 · Write what is given and what is to be found. Given: sin(x + y) = xy, an implicit equation. To find: dy/dx.
2. 2 · Write the rules. Chain rule for sin(x + y); product rule for xy; every y term gets a dy/dx.
3. 3 · Differentiate both sides. cos(x + y)(1 + dy/dx) = y + x dy/dx.
4. 4 · Expand and collect. cos(x + y) + cos(x + y) dy/dx − x dy/dx = y, so dy/dx [cos(x + y) − x] = y − cos(x + y).
5. 5 · Solve for dy/dx. dy/dx = [y − cos(x + y)]/[cos(x + y) − x].
6. 6 · Check and box the answer. Every y term carries dy/dx, and the denominator is cos(x + y) − x ≠ 0. Box the answer.

Scoring tip: Put (1 + dy/dx) in a bracket after cos(x + y). Dropping the bracket ruins the whole answer.