Implicit Differentiation

When y isn't isolated (e.g., x² + y² = 25), use implicit differentiation.

Method:

1. Differentiate both sides with respect to x

2. Every time you differentiate a y term, multiply by dy/dx (chain rule)

3. Solve for dy/dx

Example: x² + y² = 25

2x + 2y(dy/dx) = 0

dy/dx = -2x / 2y = -x/y

Example: xy + y³ = 7

(1·y + x·dy/dx) + 3y²·dy/dx = 0 (product rule on xy)

y + x·dy/dx + 3y²·dy/dx = 0

dy/dx(x + 3y²) = -y

dy/dx = -y / (x + 3y²)

AP exam: Implicit differentiation appears on free response almost every year. Often combined with finding tangent lines or related rates.

Reference:

Wikipedia: Implicit Function

https://en.wikipedia.org/wiki/Implicit_function