Implicit differentiation

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

Differentiating a relation $F(x, y) = 0$ without solving for $y$ : apply the chain rule to both sides and solve for $y'$ . The slope of a circle, an ellipse or an elliptic curve at a point comes out this way.

Formulas

$F(x, y(x)) = 0 \implies y' = -\frac{F_x}{F_y}$; $y^2 = x^3 + ax + b \implies y' = \frac{3x^2 + a}{2y}$
tangent slope of an elliptic curve

Where it shows up in computing

Elliptic curve cryptography★★★★★historicalCryptography and security
The point-doubling formula uses the slope $\frac{3x^2 + a}{2y}$ , derived over the reals and reused verbatim over finite fields.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

⊕ Cryptography and security

Elliptic curve cryptography★★★★★

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.