Implicit Differentiation AB & BC

An equation such as $x^2 + y^2 = 25$ defines $y$ as a function of $x$ without ever isolating it. Implicit differentiation takes the equation as it stands, differentiates both sides with respect to $x$, and treats every appearance of $y$ as a composition. The chain rule then forces a factor of $\frac{dy}{dx}$ onto each $y$-term, and the resulting equation is solved for that factor.

Two failure modes account for nearly every lost mark. The first is omitting $\frac{dy}{dx}$, writing $\frac{d}{dx}[y^2] = 2y$ as though $y$ were the variable of differentiation, sometimes tagging one $y$-term and forgetting another in the same line. The second is mishandling a mixed term such as $xy$ or $x^2y^3$, where the product rule has to run alongside the chain rule. Afterward the algebra has its own traps: collecting before dividing, and substituting a point only after differentiating, never before.