3.2 Implicit Differentiation

The idea

When y is defined implicitly (e.g. x² + y² = 25, not solved for y), differentiate both sides with respect to x, treating y as a function of x. By the chain rule, d/dx[y²] = 2y·(dy/dx).

Solve for dy/dx

After differentiating, collect the dy/dx terms and solve for dy/dx. For x² + y² = 25: 2x + 2y·y' = 0, so dy/dx = −x/y.

Key terms

Implicit differentiation

Differentiate both sides w.r.t. x, treating y as a function of x.

Key chain-rule step

d/dx[y²] = 2y·(dy/dx).

When to use

When y is not (easily) solved for.

Finish

Solve the resulting equation for dy/dx.

Example

x² + y² = 25 ⇒ dy/dx = −x/y.

Try an AP-style question

For x² + y² = 25, what is the slope of the tangent line at (3, 4)?

1. −4/3
2. 3/4
3. −3/4
4. 4/3