AHL 5.14 (HL)—Implicit differentiation and related rates

Implicit differentiation keeps both variables changing.

Differentiate each term with respect to x, treating y as y(x), so every y term contributes a factor dy/dx.

Example

From x²+y²=25, 2x+2y y′=0, hence y′=−x/y where y≠0.

Collect dy/dx terms, then substitute the point only after differentiating.

A vertical tangent can make the solved slope undefined even though the curve is smooth.

Related-rate workflow: write one equation connecting all changing quantities, differentiate with respect to time and only then substitute the instant's values. For a circle, $A=\pi r^2$ gives $dA/dt=2\pi r\,dr/dt$; at $r=3$ and $dr/dt=2$, $dA/dt=12\pi$ square units per unit time. For optimization, include feasible endpoints as candidates when the optimum can occur on the boundary.