Implicit differentiation

When y is not isolated — for example x² + y² = 25 — differentiate both sides with respect to x, remembering that y is a function of x, so every term containing y is multiplied by dy/dx (chain rule).

Worked examples

1. x² + y² = r²: 2x + 2y·y′ = 0 ⟹ y′ = −x/y. At (3, 4) on x² + y² = 25 the slope is −3/4, so the tangent is 3x + 4y = 25 and the normal is 4x − 3y = 0.
2. x³ + y³ = 6xy (the folium of Descartes): 3x² + 3y²y′ = 6y + 6xy′ ⟹ y′(3y² − 6x) = 6y − 3x² ⟹ y′ = (2y − x²)/(y² − 2x).
3. xy + y² = 5: y + xy′ + 2yy′ = 0 ⟹ y′ = −y/(x + 2y).
4. sin(x + y) = x ⟹ cos(x + y)(1 + y′) = 1 ⟹ y′ = sec(x + y) − 1.

Products require the product rule first: d/dx(xy) = x·y′ + y.